Find all values of x satisfying the given conditions.
-$y = \left( x ^ { 2 } - 2 x \right) ^ { 2 } - 18 \left( x ^ { 2 } - 2 x \right)$ and $y = - 45$
A) $- 1 , - 3,3,5$
B) 3,5
C) 3,15
D) $- 1 , - 3,3,15,3,5$

Question

Quizplus · Accepted Answer

Let $z = x^2 - 2x$. The equation becomes $y = z^2 - 18z$. Setting $y = -45$, we get $z^2 - 18z + 45 = 0$. Factoring gives $(z - 3)(z - 15) = 0$, so $z = 3$ or $z = 15$. Substituting back for $z$, we solve $x^2 - 2x = 3$ and $x^2 - 2x = 15$. The first gives $x^2 - 2x - 3 = 0$, which factors to $(x - 3)(x + 1) = 0$, yielding $x = 3$ or $x = -1$. The second gives $x^2 - 2x - 15 = 0$, which factors to $(x - 5)(x + 3) = 0$, yielding $x = 5$ or $x = -3$. Thus, the solutions are $x = -1, -3, 3, 5$.

Find All Values of X Satisfying the Given Conditions y=(x2−2x)2−18(x2−2x)y = \left( x ^ { 2 } - 2 x \right) ^ { 2 } - 18 \left( x ^ { 2 } - 2 x \right)y=(x2−2x)2−18(x2−2x)

Find All Values of X Satisfying the Given Conditions $y = \left( x ^ { 2 } - 2 x \right) ^ { 2 } - 18 \left( x ^ { 2 } - 2 x \right)$