Pre-Algebra Examples

$0 = 16 t^{2} + 8 t + 24$

Step 1

Rewrite the equation as $16 t^{2} + 8 t + 24 = 0$ .

$16 t^{2} + 8 t + 24 = 0$

Step 2

Factor $8$ out of $16 t^{2} + 8 t + 24$ .

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Step 2.1

Factor $8$ out of $16 t^{2}$ .

$8 (2 t^{2}) + 8 t + 24 = 0$

Step 2.2

Factor $8$ out of $8 t$ .

$8 (2 t^{2}) + 8 (t) + 24 = 0$

Step 2.3

Factor $8$ out of $24$ .

$8 (2 t^{2}) + 8 t + 8 \cdot 3 = 0$

Step 2.4

Factor $8$ out of $8 (2 t^{2}) + 8 t$ .

$8 (2 t^{2} + t) + 8 \cdot 3 = 0$

Step 2.5

Factor $8$ out of $8 (2 t^{2} + t) + 8 \cdot 3$ .

$8 (2 t^{2} + t + 3) = 0$

Step 3

Divide each term in $8 (2 t^{2} + t + 3) = 0$ by $8$ and simplify.

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Step 3.1

Divide each term in $8 (2 t^{2} + t + 3) = 0$ by $8$ .

$\frac{8 (2 t^{2} + t + 3)}{8} = \frac{0}{8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{8 (2 t^{2} + t + 3)}{8} = \frac{0}{8}$

Step 3.2.1.2

Divide $2 t^{2} + t + 3$ by $1$ .

$2 t^{2} + t + 3 = \frac{0}{8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $8$ .

$2 t^{2} + t + 3 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 2$ , $b = 1$ , and $c = 3$ into the quadratic formula and solve for $t$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$t = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$t = \frac{- 1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $3$ .

$t = \frac{- 1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 6.1.3

Subtract $24$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 6.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$t = \frac{- 1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 6.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$t = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 6.1.6

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

$t = \frac{- 1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$t = \frac{- 1 \pm i \sqrt{23}}{4}$

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

One to any power is one.

$t = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$t = \frac{- 1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $3$ .

$t = \frac{- 1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 7.1.3

Subtract $24$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 7.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$t = \frac{- 1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 7.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$t = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 7.1.6

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

$t = \frac{- 1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$t = \frac{- 1 \pm i \sqrt{23}}{4}$

Step 7.3

Change the $\pm$ to $+$ .

$t = \frac{- 1 + i \sqrt{23}}{4}$

Step 7.4

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

$t = \frac{- 1 \cdot 1 + i \sqrt{23}}{4}$

Step 7.5

Factor $- 1$ out of $i \sqrt{23}$ .

$t = \frac{- 1 \cdot 1 - (- i \sqrt{23})}{4}$

Step 7.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{23})$ .

$t = \frac{- 1 (1 - i \sqrt{23})}{4}$

Step 7.7

Move the negative in front of the fraction.

$t = - \frac{1 - i \sqrt{23}}{4}$

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

One to any power is one.

$t = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 8.1.2

Multiply $- 4 \cdot 2 \cdot 3$ .

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Step 8.1.2.1

Multiply $- 4$ by $2$ .

$t = \frac{- 1 \pm \sqrt{1 - 8 \cdot 3}}{2 \cdot 2}$

Step 8.1.2.2

Multiply $- 8$ by $3$ .

$t = \frac{- 1 \pm \sqrt{1 - 24}}{2 \cdot 2}$

Step 8.1.3

Subtract $24$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 8.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$t = \frac{- 1 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 8.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$t = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 8.1.6

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

$t = \frac{- 1 \pm i \sqrt{23}}{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$t = \frac{- 1 \pm i \sqrt{23}}{4}$

Step 8.3

Change the $\pm$ to $-$ .

$t = \frac{- 1 - i \sqrt{23}}{4}$

Step 8.4

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

$t = \frac{- 1 \cdot 1 - i \sqrt{23}}{4}$

Step 8.5

Factor $- 1$ out of $- i \sqrt{23}$ .

$t = \frac{- 1 \cdot 1 - (i \sqrt{23})}{4}$

Step 8.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{23})$ .

$t = \frac{- 1 (1 + i \sqrt{23})}{4}$

Step 8.7

Move the negative in front of the fraction.

$t = - \frac{1 + i \sqrt{23}}{4}$

Step 9

The final answer is the combination of both solutions.

$t = - \frac{1 - i \sqrt{23}}{4}, - \frac{1 + i \sqrt{23}}{4}$