Binomial Theorem
In the binomial expansion of (a + b)n, the coefficients of the 4th and 13thterms are equal to each other. Find the value of n.
Find the coefficient of xn in the expansion of (1 + x) (1 – x)n.
Write the 4th term from the end in the expansion of
If the coefficients of (r – 5)th and (2r – 1)th terms in the expansion of (1 + x)34 are equal, find the value of r.
Write the coefficient of x7y2 in the expansion of (x + 2y)9
Write the coefficient of the middle term in the expansion of (1 + x)2n.
Write the coefficient of the middle term in the expansion of (1 + x)2n.
Which term is independent of x in the expansion of ?
Write the number of terms in the expansion of
Show that the coefficient of x4 in the expansion of (1 + 2x + x2)5 is 212.
Prove that there is no term involving x6 in the expansion of .
Show that the coefficient of x4 in the expansion of
Show that the middle term in the expansion is 252.
Show that the coefficient of x-3 in the expansion is -330.
If the coefficients of x2 and x3 in the expansion of (3 + px)9 are the same then prove that
Show that the term independent of x in the expansion is -252.
Find the middle term in the expansion of
Prove that the coefficient of xn in the binomial expansion of (1 + x)2n is twice the coefficient of xn in the binomial expansion of (1 + x)2n-1.
Find the coefficient of x4 in the expansion of (1 + x)n (1 – x)n. Deduce that C2 = C0C4 – C1C3 + C2C2 – C3C1 + C4C0, where Cr stands for nCr.
If the 17th and 18th terms in the expansion of (2 + a)50 are equal, find the value of a.
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)2n are in AP, show that 2n2 – 9n + 7 = 0.
Find numerically the greatest term in the expansion of (2 + 3x)9,
Find the term independent of x in the expansion of :
Find the term independent of x in the expansion of :
Find the term independent of x in the expansion of :
Find the two middle terms in the expansion of :
Find the two middle terms in the expansion of :
Find the two middle terms in the expansion of:
Find the two middle terms in the expansion of :
Find the middle term in the expansion of :
Find the 4th term from the beginning and end in the expansion
Find the 4th term from the end in the expansion of
Write the general term in the expansion of
Show that the expansion of does not contain any term involving x-1.
Show that the expansion of does not contain any term involving x9.
Show that the term containing x3 does not exist in the expansion
Find the coefficient of
Find the coefficient of x in the expansion of (1 – 3x + 7×2) (1 – x)16.
Show that the ratio of the coefficient of x10 in the expansion of (1 – x2)10and the term independent of x in the expansion of is 1 : 32.
Find the ratio of the coefficient of x15 to the term independent of x in the
Find the coefficients of x7 and x8 in the expansion of .
Find the 13th term in the expansion of .
Find the 16th term in the expansion of .
Find the 9th term in the expansion of .
Find the 7th term in the expansion of .
Prove that
Using binomial theorem, prove that (23n – 7n -1) is divisible by 49, where n N.
Using binominal theorem, evaluate each of the following :
(i) (101)4
(ii) (98)4
(iii)(1.2)4
Prove that
Evaluate :
Evaluate :
Evaluate :
. Evaluate :
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
Using binomial theorem, expand each of the following:
By using the method of completing the square, show that the equation $2 x^{2}+x+4=0$ has no real roots.
$2 x^{2}+x+4=0$ $\Rightarrow 4 x^{2}+2 x+8=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}+2 x=-8$ $\Rightarrow(2 x)^{2}+2 \times 2 x \times...
Find the roots of the given equation: $\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$
$\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x+21=0 \quad$ (Multiplying both sides by $\left.\sqrt{3}\right)$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x=-21$ $\Rightarrow(\sqrt{3}...
Find the roots of the given equation: $\sqrt{2} x^{2}-3 x-2 \sqrt{2}=0$
$\sqrt{2} x^{2}-3 x-2 \sqrt{2}=0$ $\Rightarrow 2 x^{2}-3 \sqrt{2} x-4=0 \quad$ (Multiplying both sides by $\sqrt{2}$ ) $\Rightarrow 2 x^{2}-3 \sqrt{2} x=4$ $\Rightarrow(\sqrt{2} x)^{2}-2 \times...
Find the roots of the given equation: $\quad x^{2}-(\sqrt{2}+1) x+\sqrt{2}=0$
$\begin{array}{l} x^{2}-(\sqrt{2}+1) x+\sqrt{2}=0 \\ \Rightarrow x^{2}-(\sqrt{2}+1) x=-\sqrt{2} \\ \Rightarrow x^{2}-2 \times x...
Find the roots of the given equation: $4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0$
$\begin{array}{l} 4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0 \\ \Rightarrow 4 x^{2}+4 b x=a^{2}-b^{2} \\ \Rightarrow(2 x)^{2}+2 \times 2 x \times b+b^{2}=a^{2}-b^{2}+b^{2} \text { (Adding } b^{2}...
Find the roots of the given equation: $\frac{2}{x^{2}}-\frac{5}{x}+2=0$
$\frac{2}{x^{2}}-\frac{5}{x}+2=0$ $\Rightarrow \frac{2-5 x+2 x^{2}}{x^{2}}=0$ $\Rightarrow 2 x^{2}-5 x+2=0$ $\Rightarrow 4 x^{2}-10 x+4=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}-10...
Find the roots of the given equation: $5 x^{2}-6 x-2=0$
$\begin{array}{l} 5 x^{2}-6 x-2=0 \\ \Rightarrow 25 x^{2}-30 x-10=0 \\ \Rightarrow 25 x^{2}-30 x=10 \end{array}$ $\begin{array}{l} \Rightarrow(5 x)^{2}-2 \times 5 x \times 3+3^{2}=10+3^{2} \\...
Find the roots of the given equation: $3 x^{2}-2 x-1=0$
$3 x^{2}-2 x-1=0$ $\Rightarrow 9 x^{2}-6 x-3=0 \quad$ (Multiplying both sides by 3) $\Rightarrow 9 x^{2}-6 x=3$ $\Rightarrow(3 x)^{2}-2 \times 3 x \times 1+1^{2}=3+1^{2} \quad$ [Adding $1^{2}$ on...
Find the roots of the given equation: $7 x^{2}+3 x-4=0$
$\begin{array}{l} 7 x^{2}+3 x-4=0 \\ \Rightarrow 49 x^{2}+21 x-28=0 \end{array}$ $\Rightarrow 49 x^{2}+21 x=28$ $\Rightarrow(7 x)^{2}+2 \times 7 x \times...
Find the roots of the given equation: $8 x^{2}-14 x-15=0$
$8 x^{2}-14 x-15=0$ $\Rightarrow 16 x^{2}-28 x-30=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 16 x^{2}-28 x=30$ $\Rightarrow(4 x)^{2}-2 \times 4 x \times...
Find the roots of the given equation: $3 x^{2}-x-2=0$
$\begin{array}{l} 3 x^{2}-x-2=0 \\ \Rightarrow 9 x^{2}-3 x-6=0 \quad \text { (Multiplying both sides by 3) } \\ \Rightarrow 9 x^{2}-3 x=6 \\ \Rightarrow(3 x)^{2}-2 \times 3 x \times...
Find the roots of the given equation: $2 x^{2}+5 x-3=0$
$2 x^{2}+5 x-3=0$ $\Rightarrow 4 x^{2}+10 x-6=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}+10 x=6$ $\Rightarrow(2 x)^{2}+2 \times 2 x \times...
Find the roots of the given equation: $4 x^{2}+4 \sqrt{3} x+3=0$
$\begin{array}{l} 4 x^{2}+4 \sqrt{3} x+3=0 \\ \Rightarrow 4 x^{2}+4 \sqrt{3} x=-3 \\ \Rightarrow(2 x)^{2}+2 \times 2 x \times \sqrt{3}+(\sqrt{3})^{2}=-3+(\sqrt{3})^{2} \end{array}$ $\begin{array}{l}...
Find the roots of the given equation: $x^{2}+8 x-2=0$
$\begin{array}{l} x^{2}+8 x-2=0 \\ \Rightarrow x^{2}+8 x=2 \\ \Rightarrow x^{2}+2 \times x \times 4+4^{2}=2+4^{2} \\ \Rightarrow(x+4)^{2}=2+16=18 \\ \Rightarrow x+4=\pm \sqrt{18}=\pm 3 \sqrt{2} \\...
Find the roots of the given equation: $x^{2}-4 x+1=0$
$x^{2}-4 x+1=0$ $\Rightarrow x^{2}-4 x=-1$ $\Rightarrow x^{2}-2 \times x \times 2+2^{2}=-1+2^{2} \quad$ (Adding $2^{2}$ on both sides) $\Rightarrow(x-2)^{2}=-1+4=3$ $\Rightarrow x-2=\pm \sqrt{3}...
Find the roots of the given equation: $x^{2}-6 x+3=0$
$\begin{array}{l} x^{2}-6 x+3=0 \\ \Rightarrow x^{2}-6 x=-3 \\ \Rightarrow x^{2}-2 \times x \times 3+3^{2}=-3+3^{2} \\ \Rightarrow(x-3)^{2}=-3+9=6 \\ \Rightarrow x-3=\pm \sqrt{6} \end{array}$...
Find the roots of the given equation: $2^{2 x}-3.2^{(x+2)}+32=0$
$\begin{array}{l} 2^{2 x}-3.2^{(x+2)}+32=0 \\ \Rightarrow\left(2^{x}\right)^{2}-3.2^{x} \cdot 2^{2}+32=0 \end{array}$ Let $2^{x}$ be $y$. $\begin{array}{l} \therefore y^{2}-12 y+32=0 \\ \Rightarrow...
Find the roots of the given equation: $4^{(x+1)}+4^{(1-x)}=10$
Given: $\begin{array}{l} 4^{(x+1)}+4^{(1-x)}=10 \\ \Rightarrow 4^{x} \cdot 4+4^{1} \cdot \frac{1}{4^{4}}=10 \end{array}$ Let $4^{x}$ be $y$. $\begin{array}{l} \therefore 4 y+\frac{4}{y}=10 \\...
Find the roots of the given equation: $3^{(x+2)}+3^{-x}=10$
$\begin{array}{l} 3^{(x+2)}+3^{-x}=10 \\ 3^{x} .9+\frac{1}{3^{x}}=10 \end{array}$ Let $3^{x}$ be equal to $y$. $\begin{array}{l} \therefore 9 y+\frac{1}{y}=10 \\ \Rightarrow 9 y^{2}+1=10 y \\...
Find the roots of the given equation: $\frac{a}{(a x-1)}+\frac{b}{(b x-1)}=(a+b), x \neq \frac{1}{a}, \frac{1}{b}$
$\begin{array}{l} \frac{a}{(a x-1)}+\frac{b}{(b x-1)}=(a+b) \\ \Rightarrow\left[\frac{a}{(a x-1)}-b\right]+\left[\frac{b}{(b x-1)}-a\right]=0 \\ \Rightarrow \frac{a-b(a x-1)}{a x-1}+\frac{b-a(b...
Find the roots of the given equation: $\frac{a}{(x-b)}+\frac{b}{(x-a)}=2, x \neq b, a$
$\begin{array}{l} \frac{a}{(x-b)}+\frac{b}{(x-a)}=2 \\ \Rightarrow\left[\frac{a}{(x-b)}-1\right]+\left[\frac{b}{(x-b)}-1\right]=0 \\ \Rightarrow \frac{a-(x-b)}{x-b}+\frac{b-(x-b)}{x-b}=0...
Find the roots of the given equation: $\left(\frac{x}{x+1}\right)^{2}-5\left(\frac{x}{x+1}\right)+6=0, x \neq b, a$
$\left(\frac{x}{x+1}\right)^{2}-5\left(\frac{x}{x+1}\right)+6=0$ Putting $\frac{x}{x+1}=y$, we get: $\begin{array}{l} y^{2}-5 y+6=0 \\ \Rightarrow y^{2}-5 y+6=0 \end{array}$ $\begin{array}{l}...
Find the roots of the given equation: $\left(\frac{4 x-3}{2 x+1}\right)-10\left(\frac{2 x+1}{4 x-3}\right)=3, x \neq-\frac{1}{2}, \frac{3}{4}$
Given: $\left(\frac{4 x-3}{2 x+1}\right)-10\left(\frac{2 x+1}{4 x-3}\right)=3$ Putting $\frac{4 x-3}{2 x+1}=y$, we get: $\begin{array}{l} y-\frac{10}{y}=3 \\ \Rightarrow \frac{y^{2}-10}{y}=3 \\...
Find the roots of the given equation: $3\left(\frac{7 x+1}{5 x-3}\right)-4\left(\frac{5 x-3}{7 x+1}\right)=11, x \neq \frac{3}{5},-\frac{1}{7}$
$\begin{array}{l} 3\left(\frac{7 x+1}{5 x-3}\right)-4\left(\frac{5 x-3}{7 x+1}\right)=11, x \neq \frac{3}{5},-\frac{1}{7} \\ \Rightarrow \frac{3(7 x+1)^{2}-4(5 x-3)^{2}}{(5 x-3)(7 x+1)}=11 \\...
Find the roots of the given equation: $3\left(\frac{3 x-1}{2 x+3}\right)-2\left(\frac{2 x+3}{3 x-1}\right)=5, x \neq \frac{1}{3},-\frac{3}{2}$
$\begin{array}{l} 3\left(\frac{3 x-1}{2 x+3}\right)-2\left(\frac{2 x+3}{3 x-1}\right)=5, x \neq \frac{1}{3},-\frac{3}{2} \\ \Rightarrow \frac{3(3 x-1)^{2}-2(2 x+3)^{2}}{(2 x+3)(3 x-1)}=5 \\...
Find the roots of the given equation: $\frac{1}{x+1}+\frac{2}{x+2}=\frac{5}{x+4}, x \neq-1,-2,-4$
$\begin{array}{l} \frac{1}{x+1}+\frac{2}{x+2}=\frac{5}{x+4}, x \neq-1,-2,-4 \\ \Rightarrow \frac{x+2+2 x+2}{(x+1)(x+2)}=\frac{5}{x+4} \\ \Rightarrow \frac{3 x+4}{x^{2}+3 x+2}=\frac{5}{x+4} \\...
Find the roots of the given equation: $\frac{1}{(x-2)}+\frac{2}{(x-1)}=\frac{6}{x}, x \neq 0,1,2$
$\begin{array}{l} \frac{1}{(x-2)}+\frac{2}{(x-1)}=\frac{6}{x} \\ \Rightarrow \frac{(x-1)+2(x-2)}{(x-1)(x-2)}=\frac{6}{x} \\ \Rightarrow \frac{3 x-5}{x^{2}-3 x+2}=\frac{6}{x} \\ \Rightarrow 3 x^{2}-5...
Find the roots of the given equation: $\frac{x-1}{x-2}+\frac{x-3}{x-4}=3 \frac{1}{3}, x \neq 2,4$
$\begin{array}{l} \frac{x-1}{x-2}+\frac{x-3}{x-4}=3 \frac{1}{3}, x \neq 2,4 \\ \Rightarrow \frac{(x-1)(x-4)+(x-2)(x-3)}{(x-2)(x-4)}=\frac{10}{3} \\ \Rightarrow \frac{x^{2}-5 x+4+x^{2}-5 x+6}{x^{2}-6...
Find the roots of the given equation: $\frac{x-4}{x-5}+\frac{x-6}{x-7}=3 \frac{1}{3}, x \neq 5,7$
$\begin{array}{l} \frac{x-4}{x-5}+\frac{x-6}{x-7}=3 \frac{1}{3}, x \neq 5,7 \\ \Rightarrow \frac{(x-4)(x-7)+(x-5)(x-6)}{(x-5)(x-7)}=\frac{10}{3} \\ \Rightarrow \frac{x^{2}-11 x+28+x^{2}-11...
Find the roots of the given equation: $\quad \frac{x}{x+1}+\frac{x+1}{x}=2 \frac{4}{15}, x \neq 0,-1$
$\begin{array}{l} \frac{x}{x+1}+\frac{x+1}{x}=2 \frac{4}{15}, x \neq 0,-1 \\ \Rightarrow \frac{x^{2}+(x+1)^{2}}{x(x+1)}=\frac{34}{15} \\ \Rightarrow \frac{x^{2}+x^{2}+2 x+1}{x^{2}+x}=\frac{34}{15}...
Find the roots of the given equation: $\frac{x}{x-1}+\frac{x-1}{x}=4 \frac{1}{4}, x \neq 0,1$
$\begin{array}{l} \frac{x}{x-1}+\frac{x-1}{x}=4 \frac{1}{4}, x \neq 0,1 \\ \Rightarrow \frac{x^{2}+(x-1)^{2}}{x(x-1)}=\frac{17}{4} \\ \Rightarrow \frac{x^{2}+x^{2}-2 x+1}{x^{3}-x}=\frac{17}{4} \\...
Find the roots of the given equation: $\frac{3 x-4}{7}+\frac{7}{3 x-4}=\frac{5}{2}, x \neq \frac{4}{3}$
$\begin{array}{l} \frac{3 x-4}{7}+\frac{7}{3 x-4}=\frac{5}{2}, x \neq \frac{4}{3} \\ \Rightarrow \frac{(3 x-4)^{2}+49}{7(3 x-4)}=\frac{5}{2} \end{array}$ $\begin{array}{l} \Rightarrow \frac{9...
Find the roots of the given equation: 57. $\quad \frac{x+3}{x-2}-\frac{1-x}{x}=4 \frac{1}{4}, x \neq 2,0$
$\begin{array}{l} \frac{(x+3)}{(x-2)}-\frac{(1-x)}{x}=\frac{17}{4} \\ \Rightarrow \frac{x(x+3)-(1-x)(x-2)}{(x-2) x}=\frac{17}{4} \\ \Rightarrow \frac{x^{2}+3 x-\left(x-2-x^{2}+2 x\right)}{x^{2}-2...
Find the roots of the given equation: $\frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x}$
$\begin{array}{l} \frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x} \\ \Rightarrow \frac{1}{2 a+b+2 x}-\frac{1}{2 x}=\frac{1}{2 a}+\frac{1}{b} \\ \Rightarrow \frac{2 x-2 a-b-2 x}{2 x(2...
Find the roots of the given equation: $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}, x \neq 1,-5$
$\begin{array}{l} \frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}, x \neq 1,-5 \\ \Rightarrow \frac{x+5-x+1}{(x-1)(x+5)}=\frac{6}{7} \\ \Rightarrow \frac{6}{x^{2}+4 x-5}=\frac{6}{7} \\ \Rightarrow x^{2}+4...
Find the roots of the given equation: $\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, x \neq-1, \frac{1}{3}$
$\begin{array}{l} \frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, x \neq-1, \frac{1}{3} \\ \Rightarrow \frac{3}{x+1}-\frac{2}{3 x-1}=\frac{1}{2} \\ \Rightarrow \frac{9 x-3-2 x-2}{(x+1)(3...
Find the roots of the given equation: $\frac{4}{x}-3=\frac{5}{2 x+3}, x \neq 0,-\frac{3}{2}$
$\begin{array}{l} \frac{4}{x}-3=\frac{5}{2 x+3}, x \neq 0,-\frac{3}{2} \\ \Rightarrow \frac{4}{x}-\frac{5}{2 x+3}=3 \\ \Rightarrow \frac{8 x+12-5 x}{x(2 x+3)}=3 \\ \Rightarrow \frac{3 x+12}{2...
Find the roots of the given equation: $\frac{16}{x}-1=\frac{15}{x+1}, x \neq 0,-1$
$\begin{array}{l} \frac{16}{x}-1=\frac{15}{x+1}, x \neq 0,-1 \\ \Rightarrow \frac{16}{x}-\frac{15}{x+1}=1 \\ \Rightarrow \frac{16 x+16-15 x}{x(x+1)}=1 \\ \Rightarrow \frac{x+16}{x^{2}+x}=1 \\...
Find the roots of the given equation: $9 x^{2}-9(a+b) x+\left(2 a^{2}+5 a b+2 b^{2}\right)=0$
We write, $-9(a+b) x=-3(2 a+b) x-3(a+2 b) x$ as $\begin{array}{l} 9 x^{2} \times\left(2 a^{2}+5 a b+2 b^{2}\right)=9\left(2 a^{2}+5 a b+2 b^{2}\right) x^{2}=[-3(2 a+b) x] \times[-3(a+2 b) x] \\...
Find the roots of the given equation: $a^{2} b^{2} x^{2}+b^{2} x-a^{2} x-1=0$
$\begin{array}{l} a^{2} b^{2} x^{2}+b^{2} x-a^{2} x-1=0 \\ \Rightarrow b^{2} x\left(a^{2} x+1\right)-1\left(a^{2} x+1\right)=0 \\ \Rightarrow\left(b^{2} x-1\right)\left(a^{2} x+1\right)=0 \\...
Find the roots of the given equation: $12 a b x^{2}-\left(9 a^{2}-8 b^{2}\right) x-6 a b=0$
$\begin{array}{l} 12 a b x^{2}-\left(9 a^{2}-8 b^{2}\right) x-6 a b=0 \\ \Rightarrow 12 a b x^{2}-9 a^{2} x+8 b^{2} x-6 a b=0 \\ \Rightarrow 3 a x(4 b x-3 a)+2 b(4 b x-3 a)=0 \\ \Rightarrow(3 a x+2...
Find the roots of the given equation: $4 x^{2}-2\left(a^{2}+b^{2}\right) x+a^{2} b^{2}=0$
$\begin{array}{l} 4 x^{2}-2\left(a^{2}+b^{2}\right) x+a^{2} b^{2}=0 \\ \Rightarrow 4 x^{2}-2 a^{2} x-2 b^{2} x+a^{2} b^{2}=0 \\ \Rightarrow 2 x\left(2 x-a^{2}\right)-b^{2}\left(2 x-a^{2}\right)=0 \\...
Find the roots of the given equation: $x^{2}-4 a x-b^{2}+4 a^{2}=0$
We write, $-4 a x=-(b+2 a) x+(b-2 a) x$ as $\begin{array}{l} x^{2} \times\left(-b^{2}+4 a^{2}\right)=\left(-b^{2}+4 a^{2}\right) x^{2}=-(b+2 a) x \times(b-2 a) x \\ \therefore x^{2}-4 a x-b^{2}+4...
Find the roots of the given equation: $a b x^{2}+\left(b^{2}-a c\right) x-b c=0$
$\begin{array}{l} a b x^{2}+\left(b^{2}-a c\right) x-b c=0 \\ \Rightarrow a b x^{2}+b^{2} x-a c x-b c=0 \\ \Rightarrow b x(a x+b)-c(a x+b)=0 \\ \Rightarrow(b x-c)(a x+b)=0 \\ \Rightarrow b x-c=0...
Find the roots of the given equation: $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$
We write, $6 x=(a+4) x-(a-2) x$ as $\begin{array}{l} x^{2} \times\left[-\left(a^{2}+2 a-8\right)\right]=-\left(a^{2}+2 a-8\right) x^{2}=(a+4) x \times[-(a-2) x] \\ \therefore x^{2}+6 x-\left(a^{2}+2...
Find the roots of the given equation: $x^{2}-(2 b-1) x+\left(b^{2}-b-20\right)=0$
We write, $-(2 b-1) x=-(b-5) x-(b+4) x$ as $\begin{array}{l} x^{2} \times\left(b^{2}-b-20\right)=\left(b^{2}-b-20\right) x^{2}=[-(b-5) x] \times[-(b+4) x] \\ \therefore x^{2}-(2 b-1)...
Find the roots of the given equation: $x^{2}-2 a x-\left(4 b^{2}-a^{2}\right)=0$
We have, $-2 a x=(2 b-a) x-(2 b+a) x$ as $\begin{array}{l} x^{2} \times\left[-\left(4 b^{2}-a^{2}\right)\right]=-\left(4 b^{2}-a^{2}\right) x^{2}=(2 b-a) x \times[-(2 b+a) x] \\ \therefore x^{2}-2 a...
Find the roots of the given equation: $\quad x^{2}+5 x-\left(a^{2}+a-6\right)=0$
We write, $5 x=(a+3) x-(a-2) x$ as $\begin{array}{l} x^{2} \times\left[-\left(a^{2}+a-6\right)\right]=-\left(a^{2}+a-6\right) x^{2}=(a+3) x \times[-(a-2) x] \\ \therefore x^{2}+5...
Find the roots of the given equation: $4 x^{2}-4 a^{2} x+\left(a^{4}-b^{4}\right)=0$
We write, $-4 a^{2} x=-2\left(a^{2}+b^{2}\right) x-2\left(a^{2}-b^{2}\right) x$ as $\begin{array}{l} 4 x^{2} \times\left(a^{4}-b^{4}\right)=4\left(a^{4}-b^{4}\right)...
Find the roots of the given equation: $4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0$
We write, $4 b x=2(a+b) x-2(a-b) x$ as $\begin{array}{l} 4 x^{2} \times\left[-\left(a^{2}-b^{2}\right)\right]=-4\left(a^{2}-b^{2}\right) x^{2}=2(a+b) x \times[-2(a-b) x] \\ \therefore 4 x^{2}+4 b...
Find the roots of the given equation: $2 x^{2}+a x-a^{2}=0$
We write, $a x=2 a x-a x$ as $2 x^{2} \times\left(-a^{2}\right)=-2 a^{2} x^{2}=2 a x \times(-a x)$ $\begin{array}{l} \therefore 2 x^{2}+a x-a^{2}=0 \\ \Rightarrow 2 x^{2}+2 a x-a x-a^{2}=0 \\...
Find the roots of the given equation: $\frac{2}{x^{2}}-\frac{5}{x}+2=0$
$\begin{array}{l} \frac{2}{x^{2}}-\frac{5}{x}+2=0 \\ \Rightarrow 2-5 x+2 x^{2}=0 \\ \Rightarrow 2 x^{2}-5 x+2=0 \\ \Rightarrow 2 x^{2}-(4 x+x)+2=0 \\ \Rightarrow 2 x^{2}-4 x-x+2=0 \\ \Rightarrow 2...
Find the roots of the given equation: $10 x-\frac{1}{x}=3$
$\begin{array}{l} 10 x-\frac{1}{x}=3 \\ \Rightarrow 10 x^{2}-1=3 x \end{array}$ [Multiplying both sides by $x]$ $\begin{array}{l} \Rightarrow 10 x^{2}-3 x-1=0 \\ \Rightarrow 10 x^{2}-(5 x-2 x)-1=0...
Find the roots of the given equation: $\quad 2 x^{2}-x+\frac{1}{8}=0$
We write, $-x=-\frac{x}{2}-\frac{x}{2}$ as $2 x^{2} \times \frac{1}{8}=\frac{x^{2}}{4}=\left(-\frac{x}{2}\right) \times\left(-\frac{x}{2}\right)$ $\begin{array}{l} \therefore 2 x^{2}-x+\frac{1}{8}=0...
Find the roots of the given equation: $100 x^{2}-20 x+1=0$
We write, $-20 x=-10 x-10 x$ as $100 x^{2} \times 1=100 x^{2}=(-10 x) \times(-10 x)$ $\begin{array}{l} \therefore 100 x^{2}-20 x+1=0 \\ \Rightarrow 100 x^{2}-10 x-10 x+1=0 \end{array}$...
Find the roots of the given equation: $9 x^{2}+6 x+1=0$
$\begin{array}{l} 9 x^{2}+6 x+1=0 \\ \Rightarrow 9 x^{2}+3 x+3 x+1=0 \\ \Rightarrow 3 x(3 x+1)+1(3 x+1)=0 \\ \Rightarrow(3 x+1)(3 x+1)=0 \\ \Rightarrow 3 x+1=0 \text { or } 3 x+1=0 \\ \Rightarrow...
Find the roots of the given equation: $x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0$
$\begin{array}{l} x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0 \\ \Rightarrow x^{2}-x-\sqrt{2} x+\sqrt{2}=0 \\ \Rightarrow x(x-1)-\sqrt{2}(x-1)=0 \\ \Rightarrow(x-\sqrt{2})(x-1)=0 \\ \Rightarrow x-\sqrt{2}=0...
Find the roots of the given equation: $5 x^{2}+13 x+8=0$
We write, $13 x=5 x+8 x$ as $5 x^{2} \times 8=40 x^{2}=5 x \times 8 x$ $\begin{array}{l} \therefore 5 x^{2}+13 x+8=0 \\ \Rightarrow 5 x^{2}+5 x+8 x+8=0 \\ \Rightarrow 5 x(x+1)+8(x+1)=0 \\...
Find the roots of the given equation: $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$
We write, $7 x=5 x+2 x$ as $\sqrt{2} x^{2} \times 5 \sqrt{2}=10 x^{2}=5 x \times 2 x$ $\begin{array}{l} \therefore \sqrt{2} x^{2}+7 x+5 \sqrt{2}=0 \\ \Rightarrow \sqrt{2} x^{2}+5 x+2 x+5 \sqrt{2}=0...
Find the roots of the given equation: $\quad x^{2}+3 \sqrt{3} x-30=0$
We write, $3 \sqrt{3} x=5 \sqrt{3} x-2 \sqrt{3} x$ as $x^{2} \times(-30)=-30 x^{2}=5 \sqrt{3} x \times(-2 \sqrt{3} x)$ $\begin{array}{l} \therefore x^{2}+3 \sqrt{3} x-30=0 \\ \Rightarrow x^{2}+5...
Find the roots of the given equation: $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$
$\begin{array}{l} x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0 \\ \Rightarrow x^{2}-\sqrt{3} x-x+\sqrt{3}=0 \\ \Rightarrow x(x-\sqrt{3})-1(x-\sqrt{3})=0 \\ \Rightarrow(x-\sqrt{3})(x-1)=0 \\ \Rightarrow...
Find the roots of the given equation: $x^{2}-3 \sqrt{5} x+10=0$
We write, $-3 \sqrt{5} x=-2 \sqrt{5} x-\sqrt{5} x$ as $x^{2} \times 10=10 x^{2}=(-2 \sqrt{5} x) \times(-\sqrt{5} x)$ $\begin{array}{l} \therefore x^{2}-3 \sqrt{5} x+10=0 \\ \Rightarrow x^{2}-2...
Find the roots of the given equation: $\sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$
We write, $-2 \sqrt{2} x=-3 \sqrt{2} x+\sqrt{2} x$ as $\sqrt{3} x^{2} \times(-2 \sqrt{3})=-6 x^{2}=(-3 \sqrt{2} x) \times(\sqrt{2} x)$ $\therefore \sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$...
Find the roots of the given equation: $3 x^{2}-2 \sqrt{6 x+2}=0$
We write, $-2 \sqrt{6} x=-\sqrt{6} x$ ad $3 x^{2} \times 2=6 x^{2}=(-\sqrt{6} x) \times(-\sqrt{6} x)$ $\begin{array}{l} \therefore 3 x^{2}-2 \sqrt{6} x+2=0 \\ \Rightarrow 3 x^{2}-\sqrt{6} x-\sqrt{6}...
Find the roots of the given equation: $4 \sqrt{6} x^{2}-13 x-2 \sqrt{6}=0$
$\begin{array}{l} 4 \sqrt{6} x^{2}-13 x-2 \sqrt{6}=0 \\ \Rightarrow 4 \sqrt{6} x^{2}-16 x+3 x-2 \sqrt{6}=0 \\ \Rightarrow 4 \sqrt{2} x(\sqrt{3} x-2 \sqrt{2})+\sqrt{3}(\sqrt{3} x-2 \sqrt{2})=0 \\...
Find the roots of the given equation: $\sqrt{7} x^{2}-6 x-13 \sqrt{7}=0$
We write, $-6 x=7 x-13 x$ as $\sqrt{7} x^{2} \times(-13 \sqrt{7})=-91 x^{2}=7 x \times(-13 x)$ $\begin{array}{l} \therefore \sqrt{7} x^{2}-6 x-13 \sqrt{7}=0 \\ \Rightarrow \sqrt{7} x^{2}+7 x-13 x-13...
Find the roots of the given equation: $3 \sqrt{7} x^{2}+4 x-\sqrt{7}=0$
$\begin{array}{l} 3 \sqrt{7} x^{2}+4 x-\sqrt{7}=0 \\ \Rightarrow 3 \sqrt{7} x^{2}+7 x-3 x-\sqrt{7}=0 \\ \Rightarrow \sqrt{7} x(3 x+\sqrt{7})-1(3 x+\sqrt{7})=0 \\ \Rightarrow(3...
Find the roots of the given equation: $\sqrt{3} x^{2}+11 x+6 \sqrt{3}=0$
$\begin{array}{l} \sqrt{3} x^{2}+11 x+6 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x^{2}+9 x+2 x+6 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x(x+3 \sqrt{3})+2(x+3 \sqrt{3})=0 \end{array}$ $\begin{array}{l}...
Find the roots of the given equation: $\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$
We write: $10 x=3 x+7 x$ as $\sqrt{3} x^{2} \times 7 \sqrt{3}=21 x^{2}=3 x \times 7 x$ $\begin{array}{l} \therefore \sqrt{3} x^{2}+10 x+7 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x^{2}+3 x+7 x+7...
Find the roots of the given equation: $x^{2}+2 \sqrt{2} x-6=0$
We write: $2 \sqrt{2} x=3 \sqrt{2} x-\sqrt{2} x$ as $x^{2} \times(-6)=-6 x^{2}=3 \sqrt{2} x \times(-\sqrt{2} x)$ $\therefore x^{2}+2 \sqrt{2} x-6=0$ $\Rightarrow x^{2}+2 \sqrt{2} x-\sqrt{2} x-6=0$...
Find the roots of the given equation: $48 x^{2}-13 x-1=0$
$\begin{array}{l} 48 x^{2}-13 x-1=0 \\ \Rightarrow 48 x^{2}-(16 x-3 x)-1=0 \\ \Rightarrow 48 x^{2}-16 x+3 x-1=0 \\ \Rightarrow 16 x(3 x-1)+1(3 x-1)=0 \\ \Rightarrow(16 x+1)(3 x-1)=0 \\ \Rightarrow...
Find the roots of the given equation: $4-11 x=3 x^{2}$
$\begin{array}{l} 4-11 x=3 x^{2} \\ \Rightarrow 3 x^{2}+11 x-4=0 \\ \Rightarrow 3 x^{2}+12 x-x-4=0 \\ \Rightarrow 3 x(x+4)-1(x+4)=0 \\ \Rightarrow(x+4)(3 x-1)=0 \\ \Rightarrow x+4=0 \text { or } 3...
Find the roots of the given equation: $15 x^{2}-28=x$
$\begin{array}{l} 15 x^{2}-28=x \\ \Rightarrow 15 x^{2}-x-28=0 \\ \Rightarrow 15 x^{2}-(21 x-20 x)-28=0 \\ \Rightarrow 15 x^{2}-21 x+20 x-28=0 \end{array}$ $\begin{array}{l} \Rightarrow 3 x(5...
Find the roots of the given equation: $4 x^{2}-9 x=100$
$\begin{array}{l} 4 x^{2}-9 x=100 \\ \Rightarrow 4 x^{2}-9 x-100=0 \\ \Rightarrow 4 x^{2}-(25 x-16 x)-100=0 \\ \Rightarrow 4 x^{2}-25 x+16 x-100=0 \\ \Rightarrow x(4 x-25)+4(4 x-25)=0 \\...
Find the roots of the given equation: $3 x^{2}-2 x-1=0$
We write, $-2 x=-3 x+x$ as $3 x^{2} \times(-1)=-3 x^{2}=(-3 x) \times x$ $\begin{array}{l} \therefore 3 x^{2}-2 x-1=0 \\ \Rightarrow 3 x^{2}-3 x+x-1=0 \\ \Rightarrow 3 x(x-1)+1(x-1)=0 \\...
Find the roots of the given equation: $6 x^{2}+x-12=0$.
$\begin{array}{l} 6 x^{2}+x-12=0 \\ \Rightarrow 6 x^{2}+9 x-8 x-12=0 \\ \Rightarrow 3 x(2 x+3)-4(2 x+3)=0 \\ \Rightarrow(3 x-4)(2 x+3)=0 \\ \Rightarrow 3 x-4=0 \text { or } 2 x+3=0 \\ \Rightarrow...
Find the roots of the given equation: $6 x^{2}+11 x+3=0$.
$\begin{array}{l} 6 x^{2}+11 x+3=0 \\ \Rightarrow 6 x^{2}+9 x+2 x+3=0 \\ \Rightarrow 3 x(2 x+3)+1(2 x+3)=0 \\ \Rightarrow(3 x+1)(2 x+3)=0 \\ \Rightarrow 3 x+1=0 \text { or } 2 x+3=0 \\ \Rightarrow...
Find the roots of the given equation: $x^{2}=18 x-77$
$\begin{array}{l} x^{2}=18 x-77 \\ \Rightarrow x^{2}-18 x+77=0 \end{array}$ $\begin{array}{l} \Rightarrow x^{2}-(11 x+7 x)+77=0 \\ \Rightarrow x^{2}-11 x-7 x+77=0 \\ \Rightarrow x(x-11)-7(x-11)=0 \\...
Find the roots of the given equation: $x^{2}+12 x+35=0$
$\begin{array}{l} x^{2}+12 x+35=0 \\ \Rightarrow x^{2}+7 x+5 x+35=0 \\ \Rightarrow x(x+7)+5(x+7)=0 \\ \Rightarrow(x+5)(x+7)=0 \\ \Rightarrow x+5=0 \text { or } x+7=0 \\ \Rightarrow x=-5 \text { or }...
Find the roots of the given equation: $9 x^{2}-3 x-2=0$.
We write, $-3 x=3 x-6 x$ as $9 x^{2} \times(-2)=-18 x^{2}=3 x \times(-6 x)$ $\begin{array}{l} \therefore 9 x^{2}-3 x-2=0 \\ \Rightarrow 9 x^{2}+3 x-6 x-2=0 \\ \Rightarrow 3 x(3 x+1)-2(3 x+1)=0 \\...
Find the roots of the given equation: $x^{2}+6 x+5=0$
We write, $6 x=x+5 x$ as $x^{2} \times 5=5 x^{2}=x \times 5 x$ $\begin{array}{l} \therefore x^{2}+6 x+5=0 \\ \Rightarrow x^{2}+x-5 x+5=0 \end{array}$ $\begin{array}{l} \Rightarrow x(x+1)+5(x+1)=0 \\...
Find the roots of the given equation: $2 x^{2}+x-6=0$
We write, $x=4 x-3 x$ as $2 x^{2} \times(-6)=-12 x^{2}=4 x \times(-3 x)$ $\begin{array}{l} \therefore 2 x^{2}+x-6=0 \\ \Rightarrow 2 x^{2}+4 x-3 x-6=0 \\ \Rightarrow 2 x(x+2)-3(x+2)=0 \\...
Find the roots of the given equation: $3 x^{2}-243=0$
$\begin{array}{l} 3 x^{2}-243=0 \\ \Rightarrow 3\left(x^{2}-81\right)=0 \\ \Rightarrow(x)^{2}-(9)^{2}=0 \\ \Rightarrow(x+9)(x-9)=0 \\ \Rightarrow x+9=0 \text { or } x-9=0 \\ \Rightarrow x=-9 \text {...
Find the roots of the given equation: $4 x^{2}+5 x=0$
$\begin{array}{l} 4 x^{2}+5 x=0 \\ \Rightarrow x(4 x+5)=0 \\ \Rightarrow x=0 \text { or } 4 x+5=0 \end{array}$ $\Rightarrow x=0$ or $x=-\frac{5}{4}$ So, the roots of the given equation are 0 and...
Find the roots of the given equation: $(2 x-3)(3 x+1)=0$
$\begin{array}{l} (2 x-3)(3 x+1)=0 \\ \Rightarrow 2 x-3=0 \text { or } 3 x+1=0 \\ \Rightarrow 2 x=3 \text { or } 3 x=-1 \\ \Rightarrow x=\frac{3}{2} \text { or } x=-\frac{1}{3} \end{array}$ As a...
Find the value of $a$ and $b$ for which $x=\frac{3}{4}$ and $x=-2$ are the roots of the equation $a x^{2}+b x-6=0$
Given $\frac{3}{4}$ is a root of $a x^{2}+b x-6=0$; therefore, we have: $a \times\left(\frac{3}{4}\right)^{2}+b \times \frac{3}{4}-6=0$ $\begin{array}{l} \Rightarrow \frac{9 a}{16}+\frac{3 b}{4}=6...
Find the value of $k$ for which $x=1$ is a root of the equation $x^{2}+k x+3=0$.
It is given that $(x=1)$ is a root of $\left(x^{2}+k x+3=0\right)$. As a result, $(x=1)$ must satisfy the equation. $\begin{array}{l} \Rightarrow(1)^{2}+k \times 1+3=0 \\ \Rightarrow k+4=0 \\...
Which of the following are the roots of $3 x^{2}+2 x-1=0 ?$
$-\frac{1}{2}$
On subtracting $x=\left(-\frac{1}{2}\right)$ in the given equation, we get $\begin{array}{l} \text { L.H.S. }=3 x^{2}+2 x-1 \\ =3 \times\left(-\frac{1}{2}\right)^{2}+2...
Which of the following are the roots of $3 x^{2}+2 x-1=0 ?$
(i) $-1$
(ii) $\frac{1}{3}$
(i) $x=(-1)$ $\begin{array}{l} \text { L.H.S. }=x^{2}+2 x-1 \\ =3 \times(-1)^{2}+2 \times(-1)-1 \\ =3-2-1 \\ =0 \\ =\text { R.H.S. } \end{array}$ Thus, $(-1)$ is a root of $\left(3 x^{2}+2...
Which of the following are quadratic equation in $x$?
$\left(x+\frac{1}{x}\right)^{2}=2\left(x+\frac{1}{x}\right)+3$
$\left(x+\frac{1}{x}\right)^{2}=2\left(x+\frac{1}{x}\right)+3$ $\begin{array}{l} \Rightarrow\left(\frac{x^{2}+1}{x}\right)^{2}=2\left(\frac{x^{2}+1}{x}\right)+3 \\...
Which of the following are quadratic equation in $x$?
(i) $(x+2)^{3}=x^{3}-8$
(ii) $(2 x+3)(3 x+2)=6(x-1)(x-2)$
(i) $\begin{array}{l} (x+2)^{3}=x^{3}-8 \\ \Rightarrow x^{3}+6 x^{2}+12 x+8=x^{3}-8 \\ \Rightarrow 6 x^{2}+12 x+16=0 \end{array}$ This is of the form $a x^{2}+b x+c=0$ Hence, the given equation is a...
Which of the following are quadratic equation in $x$?
(i) $x^{2}-\frac{2}{x}=x^{2}$
(ii) $x^{2}-\frac{1}{x^{2}}=5$
(i) $\begin{array}{l} x^{2}-\frac{2}{x}=x^{2} \\ \Rightarrow x^{2}+2=x^{3} \\ \Rightarrow x^{3}-x^{2}-2=0 \end{array}$ $\left(x^{3}-x^{2}-2\right)$ is not a quadratic polynomial. $\therefore...
Which of the following are quadratic equation in $x$?
(i) $x^{2}-3 x-\sqrt{x}+4=0$
(ii) $x-\frac{6}{x}=3$
(i) $\left(x^{2}-3 x-\sqrt{x}+4\right)$ contains a term with $\sqrt{x}$, i.e., $x^{\frac{1}{2}}$, where $\frac{1}{2}$ is not a integer. Therefore, it is not a quadratic polynomial. $\therefore...
Which of the following are quadratic equation in $x$?
(i) $\sqrt{2} x^{2}+7 x+5 \sqrt{2}$
(ii) $\frac{1}{3} x^{2}+\frac{1}{5} x-2=0$
(i) Clearly, $\left(\sqrt{2} x^{2}+7 x+5 \sqrt{2}\right)$ is a quadratic polynomial. $\therefore \sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$ is a quadratic equation. (ii) Clearly, $\left(\frac{1}{3}...