Question

The line y = 2x – 3 touches the curve y = x^2 + kx + 6. Find the possible values of k.

Answer 1

The value of "k" will be a value greater than 4 or less than -8

If the line y = 2x – 3 touches the curve y = x^2 + kx + 6, this means that both equations will be equal as shown:

$2x - 3 = x^2 + kx + 6$

Rewrite in the form $ax^2+bx+c=0$

$2x - 3 = x^2 + kx + 6\\x^2+kx-2x+6+3 =0\\x^2+(k-2)x+9=0$

The roots has a distinct solution if b² - 4ac > 0

From the equation

a = 1

b = k-2

c = 9

Substitute into the formula:

$(k-2)^2-4(1)(9) > 0\\k^2-4k+4-36>0\\k^2-4k-32>0$

Factorize

$k^2+8k-4k-32 >0\\k(k+8)-4(k+8)>0\\(k-4)(k+8)>0\\k>4 and k>-8$

Hence the value of "k" will be value greater than 4 or less than -8

Learn more here: brainly.com/question/24896483