Precalculus Examples

$0.7 t - 0.4 t^{2} = 16$ , $0.4 t + 0.7 t^{2} = 64$

Step 1

Subtract $16$ from both sides of the equation.

$0.7 t - 0.4 t^{2} - 16 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 2

Factor $- 0.1$ out of $0.7 t - 0.4 t^{2} - 16$ .

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Step 2.1

Reorder $0.7 t$ and $- 0.4 t^{2}$ .

$- 0.4 t^{2} + 0.7 t - 16 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 2.2

Factor $- 0.1$ out of $- 0.4 t^{2}$ .

$- 0.1 (4 t^{2}) + 0.7 t - 16 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 2.3

Factor $- 0.1$ out of $0.7 t$ .

$- 0.1 (4 t^{2}) - 0.1 (- 7 t) - 16 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 2.4

Factor $- 0.1$ out of $- 16$ .

$- 0.1 (4 t^{2}) - 0.1 (- 7 t) - 0.1 \cdot 160 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 2.5

Factor $- 0.1$ out of $- 0.1 (4 t^{2}) - 0.1 (- 7 t)$ .

$- 0.1 (4 t^{2} - 7 t) - 0.1 \cdot 160 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 2.6

Factor $- 0.1$ out of $- 0.1 (4 t^{2} - 7 t) - 0.1 (160)$ .

$- 0.1 (4 t^{2} - 7 t + 160) = 0$

$0.4 t + 0.7 t^{2} = 64$

$- 0.1 (4 t^{2} - 7 t + 160) = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 3

Divide each term in $- 0.1 (4 t^{2} - 7 t + 160) = 0$ by $- 0.1$ and simplify.

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Step 3.1

Divide each term in $- 0.1 (4 t^{2} - 7 t + 160) = 0$ by $- 0.1$ .

$\frac{- 0.1 (4 t^{2} - 7 t + 160)}{- 0.1} = \frac{0}{- 0.1}$

$0.4 t + 0.7 t^{2} = 64$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 0.1$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 0.1 (4 t^{2} - 7 t + 160)}{- 0.1} = \frac{0}{- 0.1}$

$0.4 t + 0.7 t^{2} = 64$

Step 3.2.1.2

Divide $4 t^{2} - 7 t + 160$ by $1$ .

$4 t^{2} - 7 t + 160 = \frac{0}{- 0.1}$

$0.4 t + 0.7 t^{2} = 64$

$4 t^{2} - 7 t + 160 = \frac{0}{- 0.1}$

$0.4 t + 0.7 t^{2} = 64$

$4 t^{2} - 7 t + 160 = \frac{0}{- 0.1}$

$0.4 t + 0.7 t^{2} = 64$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 0.1$ .

$4 t^{2} - 7 t + 160 = 0$

$0.4 t + 0.7 t^{2} = 64$

$4 t^{2} - 7 t + 160 = 0$

$0.4 t + 0.7 t^{2} = 64$

$4 t^{2} - 7 t + 160 = 0$

$0.4 t + 0.7 t^{2} = 64$

Step 4

Use the quadratic formula to find the solutions.

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

$0.4 t + 0.7 t^{2} = 64$

Step 5

Substitute the values $a = 4$ , $b = - 7$ , and $c = 160$ into the quadratic formula and solve for $t$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (4 \cdot 160)}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 7$ to the power of $2$ .

$t = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot 160}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot 160$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{7 \pm \sqrt{49 - 16 \cdot 160}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.2.2

Multiply $- 16$ by $160$ .

$t = \frac{7 \pm \sqrt{49 - 2560}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm \sqrt{49 - 2560}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.3

Subtract $2560$ from $49$ .

$t = \frac{7 \pm \sqrt{- 2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.4

Rewrite $- 2511$ as $- 1 (2511)$ .

$t = \frac{7 \pm \sqrt{- 1 \cdot 2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.5

Rewrite $\sqrt{- 1 (2511)}$ as $\sqrt{- 1} \cdot \sqrt{2511}$ .

$t = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.6

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

$t = \frac{7 \pm i \cdot \sqrt{2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.7

Rewrite $2511$ as $9^{2} \cdot 31$ .

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Step 6.1.7.1

Factor $81$ out of $2511$ .

$t = \frac{7 \pm i \cdot \sqrt{81 (31)}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.7.2

Rewrite $81$ as $9^{2}$ .

$t = \frac{7 \pm i \cdot \sqrt{9^{2} \cdot 31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm i \cdot \sqrt{9^{2} \cdot 31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.8

Pull terms out from under the radical.

$t = \frac{7 \pm i \cdot (9 \sqrt{31})}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.1.9

Move $9$ to the left of $i$ .

$t = \frac{7 \pm 9 i \sqrt{31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm 9 i \sqrt{31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 6.2

Multiply $2$ by $4$ .

$t = \frac{7 \pm 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 7$ to the power of $2$ .

$t = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot 160}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot 160$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{7 \pm \sqrt{49 - 16 \cdot 160}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.2.2

Multiply $- 16$ by $160$ .

$t = \frac{7 \pm \sqrt{49 - 2560}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm \sqrt{49 - 2560}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.3

Subtract $2560$ from $49$ .

$t = \frac{7 \pm \sqrt{- 2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.4

Rewrite $- 2511$ as $- 1 (2511)$ .

$t = \frac{7 \pm \sqrt{- 1 \cdot 2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.5

Rewrite $\sqrt{- 1 (2511)}$ as $\sqrt{- 1} \cdot \sqrt{2511}$ .

$t = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.6

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

$t = \frac{7 \pm i \cdot \sqrt{2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.7

Rewrite $2511$ as $9^{2} \cdot 31$ .

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Step 7.1.7.1

Factor $81$ out of $2511$ .

$t = \frac{7 \pm i \cdot \sqrt{81 (31)}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.7.2

Rewrite $81$ as $9^{2}$ .

$t = \frac{7 \pm i \cdot \sqrt{9^{2} \cdot 31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm i \cdot \sqrt{9^{2} \cdot 31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.8

Pull terms out from under the radical.

$t = \frac{7 \pm i \cdot (9 \sqrt{31})}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.1.9

Move $9$ to the left of $i$ .

$t = \frac{7 \pm 9 i \sqrt{31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm 9 i \sqrt{31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.2

Multiply $2$ by $4$ .

$t = \frac{7 \pm 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

Step 7.3

Change the $\pm$ to $+$ .

$t = \frac{7 + 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 + 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 7$ to the power of $2$ .

$t = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot 160}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot 160$ .

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Step 8.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{7 \pm \sqrt{49 - 16 \cdot 160}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.2.2

Multiply $- 16$ by $160$ .

$t = \frac{7 \pm \sqrt{49 - 2560}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm \sqrt{49 - 2560}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.3

Subtract $2560$ from $49$ .

$t = \frac{7 \pm \sqrt{- 2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.4

Rewrite $- 2511$ as $- 1 (2511)$ .

$t = \frac{7 \pm \sqrt{- 1 \cdot 2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.5

Rewrite $\sqrt{- 1 (2511)}$ as $\sqrt{- 1} \cdot \sqrt{2511}$ .

$t = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.6

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

$t = \frac{7 \pm i \cdot \sqrt{2511}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.7

Rewrite $2511$ as $9^{2} \cdot 31$ .

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Step 8.1.7.1

Factor $81$ out of $2511$ .

$t = \frac{7 \pm i \cdot \sqrt{81 (31)}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.7.2

Rewrite $81$ as $9^{2}$ .

$t = \frac{7 \pm i \cdot \sqrt{9^{2} \cdot 31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm i \cdot \sqrt{9^{2} \cdot 31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.8

Pull terms out from under the radical.

$t = \frac{7 \pm i \cdot (9 \sqrt{31})}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.1.9

Move $9$ to the left of $i$ .

$t = \frac{7 \pm 9 i \sqrt{31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 \pm 9 i \sqrt{31}}{2 \cdot 4}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.2

Multiply $2$ by $4$ .

$t = \frac{7 \pm 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

Step 8.3

Change the $\pm$ to $-$ .

$t = \frac{7 - 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

$t = \frac{7 - 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

Step 9

The final answer is the combination of both solutions.

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$0.4 t + 0.7 t^{2} = 64$

Step 10

Subtract $64$ from both sides of the equation.

$0.4 t + 0.7 t^{2} - 64 = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 11

Factor the left side of the equation.

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Step 11.1

Factor $0.1$ out of $0.4 t + 0.7 t^{2} - 64$ .

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Step 11.1.1

Factor $0.1$ out of $0.4 t$ .

$0.1 (4 t) + 0.7 t^{2} - 64 = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 11.1.2

Factor $0.1$ out of $0.7 t^{2}$ .

$0.1 (4 t) + 0.1 (7 t^{2}) - 64 = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 11.1.3

Factor $0.1$ out of $- 64$ .

$0.1 (4 t) + 0.1 (7 t^{2}) + 0.1 (- 640) = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 11.1.4

Factor $0.1$ out of $0.1 (4 t) + 0.1 (7 t^{2})$ .

$0.1 (4 t + 7 t^{2}) + 0.1 (- 640) = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 11.1.5

Factor $0.1$ out of $0.1 (4 t + 7 t^{2}) + 0.1 (- 640)$ .

$0.1 (4 t + 7 t^{2} - 640) = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$0.1 (4 t + 7 t^{2} - 640) = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 11.2

Reorder terms.

$0.1 (7 t^{2} + 4 t - 640) = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$0.1 (7 t^{2} + 4 t - 640) = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 12

Divide each term in $0.1 (7 t^{2} + 4 t - 640) = 0$ by $0.1$ and simplify.

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Step 12.1

Divide each term in $0.1 (7 t^{2} + 4 t - 640) = 0$ by $0.1$ .

$\frac{0.1 (7 t^{2} + 4 t - 640)}{0.1} = \frac{0}{0.1}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $0.1$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{0.1 (7 t^{2} + 4 t - 640)}{0.1} = \frac{0}{0.1}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 12.2.1.2

Divide $7 t^{2} + 4 t - 640$ by $1$ .

$7 t^{2} + 4 t - 640 = \frac{0}{0.1}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$7 t^{2} + 4 t - 640 = \frac{0}{0.1}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$7 t^{2} + 4 t - 640 = \frac{0}{0.1}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Divide $0$ by $0.1$ .

$7 t^{2} + 4 t - 640 = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$7 t^{2} + 4 t - 640 = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$7 t^{2} + 4 t - 640 = 0$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 13

Use the quadratic formula to find the solutions.

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

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 14

Substitute the values $a = 7$ , $b = 4$ , and $c = - 640$ into the quadratic formula and solve for $t$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (7 \cdot - 640)}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15

Simplify.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Raise $4$ to the power of $2$ .

$t = \frac{- 4 \pm \sqrt{16 - 4 \cdot 7 \cdot - 640}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.1.2

Multiply $- 4 \cdot 7 \cdot - 640$ .

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Step 15.1.2.1

Multiply $- 4$ by $7$ .

$t = \frac{- 4 \pm \sqrt{16 - 28 \cdot - 640}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.1.2.2

Multiply $- 28$ by $- 640$ .

$t = \frac{- 4 \pm \sqrt{16 + 17920}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm \sqrt{16 + 17920}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.1.3

Add $16$ and $17920$ .

$t = \frac{- 4 \pm \sqrt{17936}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.1.4

Rewrite $17936$ as $4^{2} \cdot 1121$ .

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Step 15.1.4.1

Factor $16$ out of $17936$ .

$t = \frac{- 4 \pm \sqrt{16 (1121)}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.1.4.2

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 4 \pm \sqrt{4^{2} \cdot 1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm \sqrt{4^{2} \cdot 1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.1.5

Pull terms out from under the radical.

$t = \frac{- 4 \pm 4 \sqrt{1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm 4 \sqrt{1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.2

Multiply $2$ by $7$ .

$t = \frac{- 4 \pm 4 \sqrt{1121}}{14}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 15.3

Simplify $\frac{- 4 \pm 4 \sqrt{1121}}{14}$ .

$t = \frac{- 2 \pm 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 2 \pm 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Raise $4$ to the power of $2$ .

$t = \frac{- 4 \pm \sqrt{16 - 4 \cdot 7 \cdot - 640}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.1.2

Multiply $- 4 \cdot 7 \cdot - 640$ .

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Step 16.1.2.1

Multiply $- 4$ by $7$ .

$t = \frac{- 4 \pm \sqrt{16 - 28 \cdot - 640}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.1.2.2

Multiply $- 28$ by $- 640$ .

$t = \frac{- 4 \pm \sqrt{16 + 17920}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm \sqrt{16 + 17920}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.1.3

Add $16$ and $17920$ .

$t = \frac{- 4 \pm \sqrt{17936}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.1.4

Rewrite $17936$ as $4^{2} \cdot 1121$ .

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Step 16.1.4.1

Factor $16$ out of $17936$ .

$t = \frac{- 4 \pm \sqrt{16 (1121)}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.1.4.2

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 4 \pm \sqrt{4^{2} \cdot 1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm \sqrt{4^{2} \cdot 1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.1.5

Pull terms out from under the radical.

$t = \frac{- 4 \pm 4 \sqrt{1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm 4 \sqrt{1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.2

Multiply $2$ by $7$ .

$t = \frac{- 4 \pm 4 \sqrt{1121}}{14}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.3

Simplify $\frac{- 4 \pm 4 \sqrt{1121}}{14}$ .

$t = \frac{- 2 \pm 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.4

Change the $\pm$ to $+$ .

$t = \frac{- 2 + 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.5

Rewrite $- 2$ as $- 1 (2)$ .

$t = \frac{- 1 \cdot 2 + 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.6

Factor $- 1$ out of $2 \sqrt{1121}$ .

$t = \frac{- 1 \cdot 2 - (- 2 \sqrt{1121})}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.7

Factor $- 1$ out of $- 1 (2) - (- 2 \sqrt{1121})$ .

$t = \frac{- 1 (2 - 2 \sqrt{1121})}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 16.8

Move the negative in front of the fraction.

$t = - \frac{2 - 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = - \frac{2 - 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 17.1

Simplify the numerator.

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Step 17.1.1

Raise $4$ to the power of $2$ .

$t = \frac{- 4 \pm \sqrt{16 - 4 \cdot 7 \cdot - 640}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.1.2

Multiply $- 4 \cdot 7 \cdot - 640$ .

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Step 17.1.2.1

Multiply $- 4$ by $7$ .

$t = \frac{- 4 \pm \sqrt{16 - 28 \cdot - 640}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.1.2.2

Multiply $- 28$ by $- 640$ .

$t = \frac{- 4 \pm \sqrt{16 + 17920}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm \sqrt{16 + 17920}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.1.3

Add $16$ and $17920$ .

$t = \frac{- 4 \pm \sqrt{17936}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.1.4

Rewrite $17936$ as $4^{2} \cdot 1121$ .

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Step 17.1.4.1

Factor $16$ out of $17936$ .

$t = \frac{- 4 \pm \sqrt{16 (1121)}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.1.4.2

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 4 \pm \sqrt{4^{2} \cdot 1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm \sqrt{4^{2} \cdot 1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.1.5

Pull terms out from under the radical.

$t = \frac{- 4 \pm 4 \sqrt{1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = \frac{- 4 \pm 4 \sqrt{1121}}{2 \cdot 7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.2

Multiply $2$ by $7$ .

$t = \frac{- 4 \pm 4 \sqrt{1121}}{14}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.3

Simplify $\frac{- 4 \pm 4 \sqrt{1121}}{14}$ .

$t = \frac{- 2 \pm 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.4

Change the $\pm$ to $-$ .

$t = \frac{- 2 - 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.5

Rewrite $- 2$ as $- 1 (2)$ .

$t = \frac{- 1 \cdot 2 - 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.6

Factor $- 1$ out of $- 2 \sqrt{1121}$ .

$t = \frac{- 1 \cdot 2 - (2 \sqrt{1121})}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.7

Factor $- 1$ out of $- 1 (2) - (2 \sqrt{1121})$ .

$t = \frac{- 1 (2 + 2 \sqrt{1121})}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 17.8

Move the negative in front of the fraction.

$t = - \frac{2 + 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

$t = - \frac{2 + 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 18

The final answer is the combination of both solutions.

$t = - \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}$

$t = \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}$

Step 19

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(t - (\frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8})) (t - (- \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8})) (t - (- \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7})) = 0$

Step 20

Simplify.

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Step 20.1

To write $t$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$(t \cdot \frac{8}{8} - \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.2

Simplify terms.

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Step 20.2.1

Combine $t$ and $\frac{8}{8}$ .

$(\frac{t \cdot 8}{8} - \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.2.2

Combine the numerators over the common denominator.

$(\frac{t \cdot 8 - (7 + 9 i \sqrt{31})}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.3

Simplify the numerator.

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Step 20.3.1

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

$(\frac{8 \cdot t - (7 + 9 i \sqrt{31})}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.3.2

Apply the distributive property.

$(\frac{8 t - 1 \cdot 7 - (9 i \sqrt{31})}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.3.3

Multiply $- 1$ by $7$ .

$(\frac{8 t - 7 - (9 i \sqrt{31})}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.3.4

Multiply $9$ by $- 1$ .

$(\frac{8 t - 7 - 9 (i \sqrt{31})}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.3.5

Reorder terms.

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.4

To write $t$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t \cdot \frac{8}{8} + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.5

Simplify terms.

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Step 20.5.1

Combine $t$ and $\frac{8}{8}$ .

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{t \cdot 8}{8} + \frac{7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.5.2

Combine the numerators over the common denominator.

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{t \cdot 8 + 7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.6

Simplify the numerator.

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Step 20.6.1

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

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{8 \cdot t + 7 + 9 i \sqrt{31}}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.6.2

Reorder terms.

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{9 i \sqrt{31} + 8 t + 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.7

To write $t$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{9 i \sqrt{31} + 8 t + 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (t \cdot \frac{7}{7} + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.8

Simplify terms.

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Step 20.8.1

Combine $t$ and $\frac{7}{7}$ .

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{9 i \sqrt{31} + 8 t + 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{t \cdot 7}{7} + \frac{2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.8.2

Combine the numerators over the common denominator.

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{9 i \sqrt{31} + 8 t + 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{t \cdot 7 + 2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$

Step 20.9

Move $7$ to the left of $t$ .

$(\frac{- 9 i \sqrt{31} + 8 t - 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{9 i \sqrt{31} + 8 t + 7}{8}, \frac{7 - 9 i \sqrt{31}}{8}) (\frac{7 t + 2 - 2 \sqrt{1121}}{7}, - \frac{2 + 2 \sqrt{1121}}{7}) = 0$