Trigonometry Examples

$9 = 3 + 21 t - 16 t^{2}$

Step 1

Rewrite the equation as $3 + 21 t - 16 t^{2} = 9$ .

$3 + 21 t - 16 t^{2} = 9$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $9$ from both sides of the equation.

$3 + 21 t - 16 t^{2} - 9 = 0$

Step 2.2

Subtract $9$ from $3$ .

$21 t - 16 t^{2} - 6 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = - 16$ , $b = 21$ , and $c = - 6$ into the quadratic formula and solve for $t$ .

$\frac{- 21 \pm \sqrt{21^{2} - 4 \cdot (- 16 \cdot - 6)}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $21$ to the power of $2$ .

$t = \frac{- 21 \pm \sqrt{441 - 4 \cdot - 16 \cdot - 6}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4 \cdot - 16 \cdot - 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 21 \pm \sqrt{441 + 64 \cdot - 6}}{2 \cdot - 16}$

Step 5.1.2.2

Multiply $64$ by $- 6$ .

$t = \frac{- 21 \pm \sqrt{441 - 384}}{2 \cdot - 16}$

Step 5.1.3

Subtract $384$ from $441$ .

$t = \frac{- 21 \pm \sqrt{57}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 21 \pm \sqrt{57}}{- 32}$

Step 5.3

Simplify $\frac{- 21 \pm \sqrt{57}}{- 32}$ .

$t = \frac{21 \pm \sqrt{57}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{21 + \sqrt{57}}{32}, \frac{21 - \sqrt{57}}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{21 + \sqrt{57}}{32}, \frac{21 - \sqrt{57}}{32}$

Decimal Form:

$t = 0.89218232 \dots, 0.42031767 \dots$