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:
[tex]2x - 3 = x^2 + kx + 6[/tex]
Rewrite in the form [tex]ax^2+bx+c=0[/tex]
[tex]2x - 3 = x^2 + kx + 6\\x^2+kx-2x+6+3 =0\\x^2+(k-2)x+9=0\\[/tex]
The roots has a distinct solution if b² - 4ac > 0
From the equation
a = 1
b = k-2
c = 9
Substitute into the formula:
[tex](k-2)^2-4(1)(9) > 0\\k^2-4k+4-36>0\\k^2-4k-32>0[/tex]
Factorize
[tex]k^2+8k-4k-32 >0\\k(k+8)-4(k+8)>0\\(k-4)(k+8)>0\\k>4 and k>-8[/tex]
Hence the value of "k" will be value greater than 4 or less than -8
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