Pre-Algebra Examples

$80 = 100 \cdot ((4 + \frac{x}{2}) x)$

Step 1

Rewrite the equation as $100 \cdot ((4 + \frac{x}{2}) x) = 80$ .

$100 \cdot ((4 + \frac{x}{2}) x) = 80$

Step 2

Divide each term in $100 \cdot ((4 + \frac{x}{2}) x) = 80$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 \cdot ((4 + \frac{x}{2}) x) = 80$ by $100$ .

$\frac{100 \cdot ((4 + \frac{x}{2}) x)}{100} = \frac{80}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 \cdot ((4 + \frac{x}{2}) x)}{100} = \frac{80}{100}$

Step 2.2.1.2

Divide $(4 + \frac{x}{2}) x$ by $1$ .

$(4 + \frac{x}{2}) x = \frac{80}{100}$

Step 2.2.2

Apply the distributive property.

$4 x + \frac{x}{2} x = \frac{80}{100}$

Step 2.2.3

Multiply $\frac{x}{2} x$ .

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Step 2.2.3.1

Combine $\frac{x}{2}$ and $x$ .

$4 x + \frac{x \cdot x}{2} = \frac{80}{100}$

Step 2.2.3.2

Raise $x$ to the power of $1$ .

$4 x + \frac{x^{1} x}{2} = \frac{80}{100}$

Step 2.2.3.3

Raise $x$ to the power of $1$ .

$4 x + \frac{x^{1} x^{1}}{2} = \frac{80}{100}$

Step 2.2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x + \frac{x^{1 + 1}}{2} = \frac{80}{100}$

Step 2.2.3.5

Add $1$ and $1$ .

$4 x + \frac{x^{2}}{2} = \frac{80}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $80$ and $100$ .

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Step 2.3.1.1

Factor $20$ out of $80$ .

$4 x + \frac{x^{2}}{2} = \frac{20 (4)}{100}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $20$ out of $100$ .

$4 x + \frac{x^{2}}{2} = \frac{20 \cdot 4}{20 \cdot 5}$

Step 2.3.1.2.2

Cancel the common factor.

$4 x + \frac{x^{2}}{2} = \frac{20 \cdot 4}{20 \cdot 5}$

Step 2.3.1.2.3

Rewrite the expression.

$4 x + \frac{x^{2}}{2} = \frac{4}{5}$

Step 3

Subtract $\frac{4}{5}$ from both sides of the equation.

$4 x + \frac{x^{2}}{2} - \frac{4}{5} = 0$

Step 4

Multiply through by the least common denominator $10$ , then simplify.

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Step 4.1

Apply the distributive property.

$10 (4 x) + 10 (\frac{x^{2}}{2}) + 10 (- \frac{4}{5}) = 0$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $4$ by $10$ .

$40 x + 10 (\frac{x^{2}}{2}) + 10 (- \frac{4}{5}) = 0$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Factor $2$ out of $10$ .

$40 x + 2 (5) (\frac{x^{2}}{2}) + 10 (- \frac{4}{5}) = 0$

Step 4.2.2.2

Cancel the common factor.

$40 x + 2 \cdot (5 (\frac{x^{2}}{2})) + 10 (- \frac{4}{5}) = 0$

Step 4.2.2.3

Rewrite the expression.

$40 x + 5 x^{2} + 10 (- \frac{4}{5}) = 0$

Step 4.2.3

Cancel the common factor of $5$ .

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Step 4.2.3.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$40 x + 5 x^{2} + 10 (\frac{- 4}{5}) = 0$

Step 4.2.3.2

Factor $5$ out of $10$ .

$40 x + 5 x^{2} + 5 (2) (\frac{- 4}{5}) = 0$

Step 4.2.3.3

Cancel the common factor.

$40 x + 5 x^{2} + 5 \cdot (2 (\frac{- 4}{5})) = 0$

Step 4.2.3.4

Rewrite the expression.

$40 x + 5 x^{2} + 2 \cdot - 4 = 0$

Step 4.2.4

Multiply $2$ by $- 4$ .

$40 x + 5 x^{2} - 8 = 0$

Step 4.3

Reorder $40 x$ and $5 x^{2}$ .

$5 x^{2} + 40 x - 8 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 5$ , $b = 40$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (5 \cdot - 8)}}{2 \cdot 5}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 5 \cdot - 8}}{2 \cdot 5}$

Step 7.1.2

Multiply $- 4 \cdot 5 \cdot - 8$ .

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Step 7.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 40 \pm \sqrt{1600 - 20 \cdot - 8}}{2 \cdot 5}$

Step 7.1.2.2

Multiply $- 20$ by $- 8$ .

$x = \frac{- 40 \pm \sqrt{1600 + 160}}{2 \cdot 5}$

Step 7.1.3

Add $1600$ and $160$ .

$x = \frac{- 40 \pm \sqrt{1760}}{2 \cdot 5}$

Step 7.1.4

Rewrite $1760$ as $4^{2} \cdot 110$ .

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Step 7.1.4.1

Factor $16$ out of $1760$ .

$x = \frac{- 40 \pm \sqrt{16 (110)}}{2 \cdot 5}$

Step 7.1.4.2

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

$x = \frac{- 40 \pm \sqrt{4^{2} \cdot 110}}{2 \cdot 5}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 40 \pm 4 \sqrt{110}}{2 \cdot 5}$

Step 7.2

Multiply $2$ by $5$ .

$x = \frac{- 40 \pm 4 \sqrt{110}}{10}$

Step 7.3

Simplify $\frac{- 40 \pm 4 \sqrt{110}}{10}$ .

$x = \frac{- 20 \pm 2 \sqrt{110}}{5}$

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 $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 5 \cdot - 8}}{2 \cdot 5}$

Step 8.1.2

Multiply $- 4 \cdot 5 \cdot - 8$ .

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Step 8.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 40 \pm \sqrt{1600 - 20 \cdot - 8}}{2 \cdot 5}$

Step 8.1.2.2

Multiply $- 20$ by $- 8$ .

$x = \frac{- 40 \pm \sqrt{1600 + 160}}{2 \cdot 5}$

Step 8.1.3

Add $1600$ and $160$ .

$x = \frac{- 40 \pm \sqrt{1760}}{2 \cdot 5}$

Step 8.1.4

Rewrite $1760$ as $4^{2} \cdot 110$ .

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Step 8.1.4.1

Factor $16$ out of $1760$ .

$x = \frac{- 40 \pm \sqrt{16 (110)}}{2 \cdot 5}$

Step 8.1.4.2

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

$x = \frac{- 40 \pm \sqrt{4^{2} \cdot 110}}{2 \cdot 5}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 40 \pm 4 \sqrt{110}}{2 \cdot 5}$

Step 8.2

Multiply $2$ by $5$ .

$x = \frac{- 40 \pm 4 \sqrt{110}}{10}$

Step 8.3

Simplify $\frac{- 40 \pm 4 \sqrt{110}}{10}$ .

$x = \frac{- 20 \pm 2 \sqrt{110}}{5}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{- 20 + 2 \sqrt{110}}{5}$

Step 8.5

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{- 1 \cdot 20 + 2 \sqrt{110}}{5}$

Step 8.6

Factor $- 1$ out of $2 \sqrt{110}$ .

$x = \frac{- 1 \cdot 20 - (- 2 \sqrt{110})}{5}$

Step 8.7

Factor $- 1$ out of $- 1 (20) - (- 2 \sqrt{110})$ .

$x = \frac{- 1 (20 - 2 \sqrt{110})}{5}$

Step 8.8

Move the negative in front of the fraction.

$x = - \frac{20 - 2 \sqrt{110}}{5}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 5 \cdot - 8}}{2 \cdot 5}$

Step 9.1.2

Multiply $- 4 \cdot 5 \cdot - 8$ .

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Step 9.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 40 \pm \sqrt{1600 - 20 \cdot - 8}}{2 \cdot 5}$

Step 9.1.2.2

Multiply $- 20$ by $- 8$ .

$x = \frac{- 40 \pm \sqrt{1600 + 160}}{2 \cdot 5}$

Step 9.1.3

Add $1600$ and $160$ .

$x = \frac{- 40 \pm \sqrt{1760}}{2 \cdot 5}$

Step 9.1.4

Rewrite $1760$ as $4^{2} \cdot 110$ .

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Step 9.1.4.1

Factor $16$ out of $1760$ .

$x = \frac{- 40 \pm \sqrt{16 (110)}}{2 \cdot 5}$

Step 9.1.4.2

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

$x = \frac{- 40 \pm \sqrt{4^{2} \cdot 110}}{2 \cdot 5}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{- 40 \pm 4 \sqrt{110}}{2 \cdot 5}$

Step 9.2

Multiply $2$ by $5$ .

$x = \frac{- 40 \pm 4 \sqrt{110}}{10}$

Step 9.3

Simplify $\frac{- 40 \pm 4 \sqrt{110}}{10}$ .

$x = \frac{- 20 \pm 2 \sqrt{110}}{5}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{- 20 - 2 \sqrt{110}}{5}$

Step 9.5

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{- 1 \cdot 20 - 2 \sqrt{110}}{5}$

Step 9.6

Factor $- 1$ out of $- 2 \sqrt{110}$ .

$x = \frac{- 1 \cdot 20 - (2 \sqrt{110})}{5}$

Step 9.7

Factor $- 1$ out of $- 1 (20) - (2 \sqrt{110})$ .

$x = \frac{- 1 (20 + 2 \sqrt{110})}{5}$

Step 9.8

Move the negative in front of the fraction.

$x = - \frac{20 + 2 \sqrt{110}}{5}$

Step 10

The final answer is the combination of both solutions.

$x = - \frac{20 - 2 \sqrt{110}}{5}, - \frac{20 + 2 \sqrt{110}}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{20 - 2 \sqrt{110}}{5}, - \frac{20 + 2 \sqrt{110}}{5}$

Decimal Form:

$x = 0.19523539 \dots, - 8.19523539 \dots$