Question

For what value of k will (k - 12)x² + 2(k - 12)x + 2 = 0 give equal roots? Find the solution for that value of k.

Answer 1

What is the value of k for which (k+12) x^2+ (k+12) x-2=0 has equal roots?

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(k+12)x2+(k+12)x−2=0

There are multiple ways of arriving at the value of k given equal roots.

Method 1: We know that a quadratic equation can be written as

x2−(SumofRoots)∗x+(ProductofRoots)=0

Let’s put the equation in this form to get:

x2+(k+12)k+12∗x−2k+12=0

x2+x−2k+12=0

So Sum of Roots = -1 and hence each Root is -1/2.

Then Product of the Roots =(−1/2)∗(−1/2)=1/4

This means −2k+12=14

k+12=−8

k=−20

Method 2: When the Roots are equal, the Determinant of the equation is 0.

Given an equation of the form ax2+bx+c=0

b2−4ac=0

(k+12)2−4∗(k+12)(−2)=0

(k+12)∗(k+20)=0

k=−12or−20

But note that k cannot be -12 because we don’t get a quadratic with that value. The x^2 and x terms become 0.

So k = -20

Answer 2

Answer:

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