#### Question

Find the values of *k* for which the roots are real and equal in each of the following equation:

x^{2} - 2(5 + 2k)x + 3(7 + 10k) = 0

#### Solution

The given quadric equation is x^{2} - 2(5 + 2k)x + 3(7 + 10k) = 0, and roots are real and equal

Then find the value of *k.*

Here, a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)

As we know that D = b^{2} - 4ac

Putting the value of a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)

= (-2(5 + 2k))^{2} - 4 x (1) x 3(7 + 10k)

= 4(25 + 20k + 4k^{2}) - 12(7 + 10k)

= 100 + 80k + 16k^{2} - 84 - 120k

= 16 - 40k + 16k^{2}

The given equation will have real and equal roots, if D = 0

Thus,

16 - 40k + 16k^{2} = 0

8(2k^{2} - 5k + 2) = 0

2k^{2} - 5k + 2 = 0

Now factorizing of the above equation

2k^{2} - 5k + 2 = 0

2k2 - 4k - k + 2 = 0

2k(k - 2) - 1(k - 2) = 0

(k - 2)(2k - 1) = 0

So, either

k - 2 = 0

k = 2

Or

2k - 1 = 0

2k = 1

k = 1/2

Therefore, the value of k = 2, 1/2.