Kx²-4x (k+1) + 8k=7 has two equal roots, find the value of k

k = 4.

Step-by-step explanation:

For equal roots the discriminant b^2 - 4ac = 0, so for

kx² - 4x (k+1) + 8k - 7 = 0

b^2 - 4ac = (-4 (k+1) ^2 - 4 * k (8k - 7) = 0

16 (k^2 + 2k + 1) - 32k^2 + 28k = 0

- 16k^2 + 60k + 16 = 0

-4k^2 + 15k + 4 = 0

4k^2 - 15k - 4 = 0

(4k + 1) (k - 4) = 0

So k = - 0.25 or 4.

Check by substituting k = 4 into the original equation:

4x^2 - 4 * (5) x + 32 - 7 = 0

4x^2 - 20x + 25 = 0

(2x - 5) ^2 = 0.

This has equal roots so k = 4 is a correct value.

For k = - 0.25:

-0.25x^2 - 4*-0.25 (-0.25+1) x + 8*-0.25 - 7

-0.25x^2 - 1.25x - 9 = 0

This has complex roots:

b^2 - 4ac = (-1.25) ^2 - 4 * - 0.25 * - 9

= - 7.44 so this hasn't got equal roots.

Answer is k = 4.