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### Section 1.1 : Review : Functions

12. Determine all the roots of $$h\left( t \right) = 18 - 3t - 2{t^2}$$.

Show Solution

Set the function equal to zero and because the left side will not factor we’ll need to use the quadratic formula to find the roots of the function.

$18 - 3t - 2{t^2} = 0$ $t = \frac{{3 \pm \sqrt {{{\left( { - 3} \right)}^2} - 4\left( { - 2} \right)\left( {18} \right)} }}{{2\left( { - 2} \right)}} = \frac{{3 \pm \sqrt {153} }}{{ - 4}} = \frac{{3 \pm \sqrt {\left( 9 \right)\left( {17} \right)} }}{{ - 4}} = \frac{{3 \pm 3\sqrt {17} }}{{ - 4}} = - \frac{3}{4}\left( {1 \pm \sqrt {17} } \right)$

So, the quadratic formula gives the following two roots of the function,

$- \frac{3}{4}\left( {1 + \sqrt {17} } \right) = - 3.842329\hspace{0.25in}\hspace{0.25in} - \frac{3}{4}\left( {1 - \sqrt {17} } \right) = 2.342329$