Question

question 7 · 1 point for which value(s) of x does f(x)=8x³ + 5x² - 10x + 15 have a tangent line with a slope of -7? if there is more than one answer, give all of the x - values separated by commas, e.g. if f(x) has a tangent line of slope -7 at x = 3 and x = 5 enter 3, 5. provide your answer below:

Explanation:

Step1: Differentiate $f(x)$

$f'(x)=24x^{2}+10x - 10$

Step2: Set $f'(x)=-7$

$24x^{2}+10x - 10=-7$
$24x^{2}+10x - 3 = 0$

Step3: Solve quadratic equation

Using quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for $ax^{2}+bx + c = 0$. Here $a = 24$, $b = 10$, $c=-3$.
$x=\frac{-10\pm\sqrt{10^{2}-4\times24\times(-3)}}{2\times24}=\frac{-10\pm\sqrt{100 + 288}}{48}=\frac{-10\pm\sqrt{388}}{48}=\frac{-10\pm2\sqrt{97}}{48}=\frac{-5\pm\sqrt{97}}{24}$

Answer:

$\frac{-5+\sqrt{97}}{24},\frac{-5 - \sqrt{97}}{24}$