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

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

(i) 2{{x}^{2}}+kx+3=0

Solution:

Given,

2{{x}^{2}}+kx+3=0

It’s of the form of a{{x}^{2}}+bx+c=0

Where, a=2,b=k,c=3

For the given quadratic equation to have real roots D={{\left( k \right)}^{2}}-4\left( 3 \right)\left( 2 \right)\ge 0

D={{\left( k \right)}^{2}}-4\left( 3 \right)\left( 2 \right)\ge 0

⇒ {{k}^{2}}-24\ge 0

⇒ {{k}^{2}}\ge 24

⇒ k\ge 2\sqrt{6} and k\le -2\sqrt{6} [After taking square root on both sides]

The value of k can be represented as (\infty ,2\sqrt{6}]U[-2\sqrt{6},-\infty )

(ii) kx\left( x-2 \right)+6=0

Solution:

Given,

kx\left( x-2 \right)+6=0

It can be rewritten as,

k{{x}^{2}}-2kx+6=0

It’s of the form of a{{x}^{2}}+bx+c=0

Where, a=k,b=-2k,c=6

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

D={{\left( -2k \right)}^{2}}-4\left( k \right)\left( 6 \right)\ge 0

⇒ 4{{k}^{2}}-24k\ge 0

⇒ 4k\left( k-6 \right)\ge 0

⇒ k\ge 0 and k\ge 6

⇒ k\ge 6

The value of k should be greater than or equal to 6 to have real roots.

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