$12 a b x^{2}-\left(9 a^{2}-8 b^{2}\right) x-6 a b=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=12 a b, B=-\left(9 a^{2}-8 b^{2}\right)$ and $C=-6 a b$ Discriminant $D$ is given by:...
The given equation is $a^{2} b^{2} x^{2}-\left(4 b^{4}-3 a^{4}\right) x-12 a^{2} b^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=a^{2} b^{2}, B=-\left(4 b^{4}-3 a^{4}\right)$ and $c=-12...
$3 a^{2} x^{2}+8 a b x+4 b^{2}=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=3 a^{2}, B=8 a b$ and $C=4 b^{2}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(B^{2}-4 A C\right) \\...
The given equation is $x^{2}-(2 b-1) x+\left(b^{2}-b-20\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-(2 b-1)$ and $C=b^{2}-b-20$ $\therefore$ Discriminant, $D=B^{2}-4 A C=[-(2...
The given equation is $4 x^{2}-4 b x-\left(a^{2}-b^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=4 b$ and $C=-\left(a^{2}-b^{2}\right)$ $\therefore$ Discriminant, $D=B^{2}-4 A...
The given equation is $4 x^{2}-4 a^{2} x+\left(a^{4}-b^{4}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=-4 a^{2}$ and $C=a^{4}-b^{4}$ $\therefore$ Discriminant, $B^{2}-4 A...
The given equation is $x^{2}-4 a x-b^{2}+4 a^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-4 a$ and $C=-b^{2}+4 a^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-4 a)^{2}-4 \times 1...
The given equation is $x^{2}+5 x-\left(a^{2}+a-6\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=5$ and $C=-\left(a^{2}+a-8\right)$ $\therefore$ Discriminant, $D=$ $\begin{array}{l}...
The given equation is $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=6$ and $C=-\left(a^{2}+2 a-8\right)$ $\therefore$ Discriminant, $D=$...
The given equation is $x^{2}-2 a x-\left(4 b^{2}-a^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-2 a$ and $C=-\left(4 b^{2}-a^{2}\right)$ $\therefore$ Discriminant, $B^{2}-4 A...
Given: $x^{2}-2 a x+\left(a^{2}-b^{2}\right)=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=1, B=-2 a$ and $C=\left(a^{2}-b^{2}\right)$ Discriminant $D$ is given by: $\begin{array}{l}...
The given equation is $36 x^{2}-12 a x+\left(a^{2}-b^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=36, B=-12 a$ and $C=a^{2}-b^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-12...
The given equation is $\begin{array}{l} \frac{m}{n} x^{2} \frac{n}{m}=1-2 x \\ \Rightarrow \frac{m^{2} x^{2}+n^{2}}{m n}=1-2 x \\ \Rightarrow m^{2} x^{2}+n^{2}=m n-2 m n x \\ \Rightarrow m^{2}...
The given equation is $\begin{array}{l} x-\frac{1}{x}=3, x \neq 0 \\ \Rightarrow \frac{x^{2}-1}{x}=3 \\ \Rightarrow x^{2}-1=3 x \\ \Rightarrow x^{2}-3 x-1=0 \end{array}$ This equation is of the form...
The given equation is $\begin{array}{l} \frac{1}{x}-\frac{1}{x-2}=3, x \neq 0,2 \\ \Rightarrow \frac{x-2-x}{x(x-2)}=3 \\ \Rightarrow \frac{-2}{x^{2}-2 x}=3 \\ \Rightarrow-2=3 x^{2}-6 x \\...
The given equation is $\begin{array}{l} x+\frac{1}{x}=3, x \neq 0 \\ \Rightarrow \frac{x^{2}+1}{x}=3 \\ \Rightarrow x^{2}+1=3 x \\ \Rightarrow x^{2}-3 x+1=0 \end{array}$ This equation is of the form...
The given equation is $3 x^{2}-2 x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=(-2)^{2}-4 \times 3 \times 2=4-24=-20<0$ Hence,...
The given equation is $2 x^{2}+5 \sqrt{3} x+6=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=5 \sqrt{3}$ and $c=6$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(5 \sqrt{3})^{2}-4 \times 2...
The given equation is $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=-(\sqrt{3}+1)$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a...
The given equation is $2 x^{2}+a x-a^{2}=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=2, B=a$ and $C=-a^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=a^{2}-4 \times 2 \times-a^{2}=a^{2}+8...
The given equation is $x^{2}+x+2=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=1$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=1^{2}-4 \times 1 \times 2=1-8=-7<0$ Hence, the given...
The given equation is $2 \sqrt{3} x^{2}-5 x+\sqrt{3}=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2 \sqrt{3}, b=-5$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-5)^{2}-4 \times...
The given equation is $3 x^{2}-2 \sqrt{6} x+2=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2 \sqrt{6}$ and $c=2$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{6})^{2}-4 \times 3...
The given equation is $4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=4 \sqrt{3}, b=5$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=5^{2}-4 \times...
The given equation is $2 x^{2}+6 \sqrt{3} x-60=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=6 \sqrt{3}$ and $c=-60$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(6 \sqrt{3})^{2}-4 \times 2...
The given equation is $\sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=\sqrt{3}, b=-2 \sqrt{2}$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a...
Given: $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=\sqrt{3}, b=10$ and $c=-8 \sqrt{3}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a...
The given equation is $\sqrt{2} x^{2}+7+5 \sqrt{2}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=\sqrt{2}, b=7$ and $c=5 \sqrt{2}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(7)^{2}-4 \times...
The given equation is $2 x^{2}-2 \sqrt{2} x+1=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=-2 \sqrt{2}$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{2})^{2}-4 \times 2...
Given: $\begin{array}{l} 15 x^{2}-28=x \\ \Rightarrow 15 x^{2}-x-28=0 \end{array}$ On comparing it with $a x^{2}+b x+c=0$, we get; $a=25, b=-1$ and $c=-28$ Discriminant $D$ is given by:...
Given: $\begin{array}{l} 16 x^{2}+24 x+1 \\ \Rightarrow 16 x^{2}-24 x-1=0 \end{array}$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=16, b=-24$ and $c=-1$ Discriminant $D$ is given by:...
Given: $25 x^{2}+30 x+7=0$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=25, b=30$ and $c=7$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =30^{2}-4 \times 25...
The given equation is $2 x^{2}+x-4=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=1$ and $c=-4$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(1)^{2}-4 \times 2 \times(-4)=1+32=33>0$ So, the...
Given: $x^{2}-6 x+4=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-6$ and $c=4$ Discriminant $D$ is given by: $ \begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 1...
Given: $x^{2}-4 x-1=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-4$ and $c=-1$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-4)^{2}-4 \times 1...
$\begin{array}{l} 1-x=2 x^{2} \\ \Rightarrow 2 x^{2}+x-1=0 \end{array}$ Here, $\begin{array}{l} a=2 \\ b=1, \\ c=-1 \end{array}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\...
$\begin{array}{l} \Rightarrow 2 x^{2}-3 x+1=0 \end{array}$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=-3$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-3)^{2}-4 \times 2 \times...
Here, $\begin{aligned} a &=\sqrt{3} \\ b &=2 \sqrt{2} \\ c &=-2 \sqrt{3} \end{aligned}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\ =(2 \sqrt{2})^{2}-4 \times...
Here, $\begin{array}{l} a=2 \\ b=-5 \sqrt{2} \\ c=4 \end{array}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-5 \sqrt{2})^{2}-4 \times 2 \times 4 \\ =(25 \times 2)-32 \\...
Here, $a=3$ $b=-2$ $c=8$ Discriminant $D$ is given by: $D=b^{2}-4 a c$ $=(-2)^{2}-4 \times 3 \times 8$ $=4-96$ $=-92$
Here, $\begin{array}{l} a=2 \\ b=-7 \\ c=6 \end{array}$ Discriminant $D$ is diven by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-7)^{2}-4 \times 2 \times 6 \\ =49-48 \\ =1 \end{array}$