Trigonometry Examples

$32 = - 64 t^{2} + 64 t + 4$

Step 1

Rewrite the equation as $- 64 t^{2} + 64 t + 4 = 32$ .

$- 64 t^{2} + 64 t + 4 = 32$

Step 2

Subtract $32$ from both sides of the equation.

$- 64 t^{2} + 64 t + 4 - 32 = 0$

Step 3

Subtract $32$ from $4$ .

$- 64 t^{2} + 64 t - 28 = 0$

Step 4

Factor $- 4$ out of $- 64 t^{2} + 64 t - 28$ .

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Step 4.1

Factor $- 4$ out of $- 64 t^{2}$ .

$- 4 (16 t^{2}) + 64 t - 28 = 0$

Step 4.2

Factor $- 4$ out of $64 t$ .

$- 4 (16 t^{2}) - 4 (- 16 t) - 28 = 0$

Step 4.3

Factor $- 4$ out of $- 28$ .

$- 4 (16 t^{2}) - 4 (- 16 t) - 4 \cdot 7 = 0$

Step 4.4

Factor $- 4$ out of $- 4 (16 t^{2}) - 4 (- 16 t)$ .

$- 4 (16 t^{2} - 16 t) - 4 \cdot 7 = 0$

Step 4.5

Factor $- 4$ out of $- 4 (16 t^{2} - 16 t) - 4 (7)$ .

$- 4 (16 t^{2} - 16 t + 7) = 0$

Step 5

Divide each term in $- 4 (16 t^{2} - 16 t + 7) = 0$ by $- 4$ and simplify.

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Step 5.1

Divide each term in $- 4 (16 t^{2} - 16 t + 7) = 0$ by $- 4$ .

$\frac{- 4 (16 t^{2} - 16 t + 7)}{- 4} = \frac{0}{- 4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 4 (16 t^{2} - 16 t + 7)}{- 4} = \frac{0}{- 4}$

Step 5.2.1.2

Divide $16 t^{2} - 16 t + 7$ by $1$ .

$16 t^{2} - 16 t + 7 = \frac{0}{- 4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 4$ .

$16 t^{2} - 16 t + 7 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 16$ , $b = - 16$ , and $c = 7$ into the quadratic formula and solve for $t$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (16 \cdot 7)}}{2 \cdot 16}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 16$ to the power of $2$ .

$t = \frac{16 \pm \sqrt{256 - 4 \cdot 16 \cdot 7}}{2 \cdot 16}$

Step 8.1.2

Multiply $- 4 \cdot 16 \cdot 7$ .

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Step 8.1.2.1

Multiply $- 4$ by $16$ .

$t = \frac{16 \pm \sqrt{256 - 64 \cdot 7}}{2 \cdot 16}$

Step 8.1.2.2

Multiply $- 64$ by $7$ .

$t = \frac{16 \pm \sqrt{256 - 448}}{2 \cdot 16}$

Step 8.1.3

Subtract $448$ from $256$ .

$t = \frac{16 \pm \sqrt{- 192}}{2 \cdot 16}$

Step 8.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$t = \frac{16 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 16}$

Step 8.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$t = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 16}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{16 \pm i \cdot \sqrt{192}}{2 \cdot 16}$

Step 8.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 8.1.7.1

Factor $64$ out of $192$ .

$t = \frac{16 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 16}$

Step 8.1.7.2

Rewrite $64$ as $8^{2}$ .

$t = \frac{16 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 16}$

Step 8.1.8

Pull terms out from under the radical.

$t = \frac{16 \pm i \cdot (8 \sqrt{3})}{2 \cdot 16}$

Step 8.1.9

Move $8$ to the left of $i$ .

$t = \frac{16 \pm 8 i \sqrt{3}}{2 \cdot 16}$

Step 8.2

Multiply $2$ by $16$ .

$t = \frac{16 \pm 8 i \sqrt{3}}{32}$

Step 8.3

Simplify $\frac{16 \pm 8 i \sqrt{3}}{32}$ .

$t = \frac{2 \pm i \sqrt{3}}{4}$

Step 9

The final answer is the combination of both solutions.

$t = \frac{2 + i \sqrt{3}}{4}, \frac{2 - i \sqrt{3}}{4}$