Answer
If $t(x)=10$, then $t=1.6$ or $t=-1$
If $t(x)=15$, then $t\approx1.940$ or $t\approx-1.340$
Work Step by Step
$t(x)=5x^{2}-3x+2$
This problem is much harder, because it requires us to work backwards for the $x$ values. What you must do is solve the equation as $5x^{2}-3x+2=(10,15)$
So to set up, we will do $5x^{2}-3x+2=10$ and solve.
$5x^{2}-3x+2=10$
$5x^{2}-3x-8=0$ (Subtract 10 from both sides)
Use the Quadratic Formula to solve both roots:$(\frac{-b(+/-)\sqrt {b^{2}-4ac}}{2a})$
Positive Root:
$\frac{(-(-3)+\sqrt {3^{2}-4(5)(-8)}}{2(5)}$ (Substitute)
$\frac{3+\sqrt {169}}{10}=1.6$ (Simplify and solve)
Negative Root:
$\frac{(-(-3)-\sqrt {3^{2}-4(5)(-8)}}{2(5)}$ (Substitute)
$\frac{3-\sqrt {169}}{10}=-1$ (Simplify and solve)
You can solve the other part of the problem in the same fashion.
$5x^{2}-3x+2=15$
$5x^{2}-3x-13=0$
Positive Root:
$\frac{(-(-3)+\sqrt {3^{2}-4(5)(-13)}}{2(5)}$
$\frac{3+\sqrt {269}}{10}\approx1.940$
Negative Root:
$\frac{(-(-3)-\sqrt {3^{2}-4(5)(-13)}}{2(5)}$
$\frac{3-\sqrt {269}}{10}\approx$-1.340
