Make an appropriate substitution in order to obtain a quadratic equation. Find all complex values for x.
-$$\left( x ^ { 2 } - 2 x \right) ^ { 2 } - 11 \left( x ^ { 2 } - 2 x \right) + 24 = 0$$
A) $x = - 2 , - 1,8,3,4,3$
B) $x = 4,3$
C) $x = 8,3$
D) $x = - 2 , - 1,4,3$

Question

Quizplus · Accepted Answer

Let $y = x^2 - 2x$. The equation becomes $y^2 - 11y + 24 = 0$, which factors to $(y - 3)(y - 8) = 0$. Thus, $y = 3$ or $y = 8$. Substituting back, $x^2 - 2x = 3$ and $x^2 - 2x = 8$. Solving these gives $x = -1, 3$ for the first and $x = -2, 4$ for the second, totaling four solutions: $x = -2, -1, 3, 4$.

Make an Appropriate Substitution in Order to Obtain a Quadratic (x2−2x)2−11(x2−2x)+24=0\left( x ^ { 2 } - 2 x \right) ^ { 2 } - 11 \left( x ^ { 2 } - 2 x \right) + 24 = 0(x2−2x)2−11(x2−2x)+24=0

Make an Appropriate Substitution in Order to Obtain a Quadratic $\left( x ^ { 2 } - 2 x \right) ^ { 2 } - 11 \left( x ^ { 2 } - 2 x \right) + 24 = 0$