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4. Find the values of k for which the following equations have real and equal roots
Class 10 | ICSE | Maths | Quadratic Equations | RD Sharma
4. Find the values of k for which the following equations have real and equal roots

Quadratic is that type of problem which deals with a variable multiplied by itself – an operation known also as squaring.

(i) {{x}^{2}}-2\left( k+1 \right)x+{{k}^{2}}=0

Solution:

Given,

{{x}^{2}}-2\left( k+1 \right)x+{{k}^{2}}=0

It’s of the form of a{{x}^{2}}+bx+c=0

Where, a=1,b=-2\left( k+1 \right),c={{k}^{2}}

For the given quadratic equation to have real roots D={{b}^{2}}-4ac=0

D={{\left( -2\left( k+1 \right) \right)}^{2}}-4\left( 1 \right)\left( {{k}^{2}} \right)=0

⇒ 4{{k}^{2}}+8k+4-4{{k}^{2}}=0

⇒ 8k+4=0

⇒ k=-4/8

⇒ k=-1/2

The value of k should -1/2 to have real and equal roots.

(ii){{k}^{2}}{{x}^{2}}-2\left( 2k-1 \right)x+4=0

Solution:

Given,

{{k}^{2}}{{x}^{2}}-2\left( 2k-1 \right)x+4=0

It’s of the form of a{{x}^{2}}+bx+c=0

Where, a={{k}^{2}},b=-2\left( 2k-1 \right),c=4

For the given quadratic equation to have real roots D={{b}^{2}}-4ac=0

D={{\left( -2\left( 2k-1 \right) \right)}^{2}}-4\left( 4 \right)\left( {{k}^{2}} \right)=0

⇒ 4{{k}^{2}}-4k+1-4{{k}^{2}}=0 [dividing by 4 both sides]

⇒ -4k+1=0

⇒ k=1/4

The value of k should 1/4 to have real and equal roots.

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