For what value of K will the quadratic equation(K-12) x squared+2(K-12)x+2=0 give equal roots? Also, find the solution for that value of K?

For a quadratic equation to have equal roots, the discriminant must be equal to zero.

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac.

In this case, the quadratic equation is (K-12)x^2 + 2(K-12)x + 2 = 0.

So, a = K-12, b = 2(K-12), and c = 2.

The discriminant is:
(2(K-12))^2 - 4(K-12)(2) = 0
4(K^2 - 24K + 144) - 8(K-12) = 0
4K^2 - 96K + 576 - 8K + 96 = 0
4K^2 - 104K + 672 = 0

Now, set the discriminant equal to zero:
104^2 - 4(4)(672) = 0
10816 - 10752 = 64

Therefore, the discriminant is equal to zero when K = 16.

For K = 16, the quadratic equation becomes:
(16-12)x^2 + 2(16-12)x + 2 = 0
4x^2 + 8x + 2 = 0

The roots of this equation can be found using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-8 ± √(64 - 32)) / 8
x = (-8 ± √32) / 8
x = (-8 ± 4√2) / 8
x = -1 ± 0.5√2

Therefore, for K = 16, the quadratic equation (K-12)x^2 + 2(K-12)x + 2 = 0 will have equal roots of -1 ± 0.5√2.