Solve the equation by using substitution.
-$$3 \left( t ^ { 2 } - 9 \right) ^ { 2 } + 16 \left( t ^ { 2 } - 9 \right) = - 5$$
A) $\left\{ \pm \frac { \sqrt { 78 } } { 3 } , \pm 2 \right\}$
B) $\left\{ \pm \frac { \sqrt { 78 } } { 3 } i , \pm i 2 \right\}$
C) $\left\{ - \frac { 1 } { 3 } , - 5 \right\}$
D) $\left\{ \pm \frac { 1 } { 3 } , \pm 5 \right\}$

Question

Quizplus · Accepted Answer

Let $u = t^2 - 9$. The equation becomes $3u^2 + 16u + 5 = 0$. Solving this quadratic equation for $u$ gives $u = -\frac{5}{3}$ or $u = -1$. Substituting back for $t^2 - 9$, we get $t^2 - 9 = -\frac{5}{3}$ or $t^2 - 9 = -1$, leading to the solutions for $t$ as $\pm \frac{\sqrt{78}}{3}$ and $\pm 2$.

Solve the Equation by Using Substitution 3(t2−9)2+16(t2−9)=−53 \left( t ^ { 2 } - 9 \right) ^ { 2 } + 16 \left( t ^ { 2 } - 9 \right) = - 53(t2−9)2+16(t2−9)=−5

Solve the Equation by Using Substitution $3 \left( t ^ { 2 } - 9 \right) ^ { 2 } + 16 \left( t ^ { 2 } - 9 \right) = - 5$