Pre-Algebra Examples

$(2) (h + 8) (h) = 10$

Step 1

Divide each term in $2 (h + 8) h = 10$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 (h + 8) h = 10$ by $2$ .

$\frac{2 (h + 8) h}{2} = \frac{10}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 (h + 8) h}{2} = \frac{10}{2}$

Step 1.2.1.2

Divide $(h + 8) h$ by $1$ .

$(h + 8) h = \frac{10}{2}$

Step 1.2.2

Apply the distributive property.

$h \cdot h + 8 h = \frac{10}{2}$

Step 1.2.3

Multiply $h$ by $h$ .

$h^{2} + 8 h = \frac{10}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $10$ by $2$ .

$h^{2} + 8 h = 5$

Step 2

Subtract $5$ from both sides of the equation.

$h^{2} + 8 h - 5 = 0$

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 $a = 1$ , $b = 8$ , and $c = - 5$ into the quadratic formula and solve for $h$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $8$ to the power of $2$ .

$h = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$h = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 5}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 5$ .

$h = \frac{- 8 \pm \sqrt{64 + 20}}{2 \cdot 1}$

Step 5.1.3

Add $64$ and $20$ .

$h = \frac{- 8 \pm \sqrt{84}}{2 \cdot 1}$

Step 5.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 5.1.4.1

Factor $4$ out of $84$ .

$h = \frac{- 8 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 5.1.4.2

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

$h = \frac{- 8 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$h = \frac{- 8 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$h = \frac{- 8 \pm 2 \sqrt{21}}{2}$

Step 5.3

Simplify $\frac{- 8 \pm 2 \sqrt{21}}{2}$ .

$h = - 4 \pm \sqrt{21}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $8$ to the power of $2$ .

$h = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$h = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 5$ .

$h = \frac{- 8 \pm \sqrt{64 + 20}}{2 \cdot 1}$

Step 6.1.3

Add $64$ and $20$ .

$h = \frac{- 8 \pm \sqrt{84}}{2 \cdot 1}$

Step 6.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 6.1.4.1

Factor $4$ out of $84$ .

$h = \frac{- 8 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 6.1.4.2

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

$h = \frac{- 8 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$h = \frac{- 8 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$h = \frac{- 8 \pm 2 \sqrt{21}}{2}$

Step 6.3

Simplify $\frac{- 8 \pm 2 \sqrt{21}}{2}$ .

$h = - 4 \pm \sqrt{21}$

Step 6.4

Change the $\pm$ to $+$ .

$h = - 4 + \sqrt{21}$

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 $8$ to the power of $2$ .

$h = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$h = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 5}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 5$ .

$h = \frac{- 8 \pm \sqrt{64 + 20}}{2 \cdot 1}$

Step 7.1.3

Add $64$ and $20$ .

$h = \frac{- 8 \pm \sqrt{84}}{2 \cdot 1}$

Step 7.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 7.1.4.1

Factor $4$ out of $84$ .

$h = \frac{- 8 \pm \sqrt{4 (21)}}{2 \cdot 1}$

Step 7.1.4.2

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

$h = \frac{- 8 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$h = \frac{- 8 \pm 2 \sqrt{21}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$h = \frac{- 8 \pm 2 \sqrt{21}}{2}$

Step 7.3

Simplify $\frac{- 8 \pm 2 \sqrt{21}}{2}$ .

$h = - 4 \pm \sqrt{21}$

Step 7.4

Change the $\pm$ to $-$ .

$h = - 4 - \sqrt{21}$

Step 8

The final answer is the combination of both solutions.

$h = - 4 + \sqrt{21}, - 4 - \sqrt{21}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$h = - 4 + \sqrt{21}, - 4 - \sqrt{21}$

Decimal Form:

$h = 0.58257569 \dots, - 8.58257569 \dots$