Pre-Algebra Examples

$(3 x + 15) (4 x) = 180$

Step 1

Remove parentheses.

$(3 x + 15) \cdot 4 x = 180$

Step 2

Divide each term in $(3 x + 15) \cdot (4 x) = 180$ by $4$ and simplify.

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Step 2.1

Divide each term in $(3 x + 15) \cdot (4 x) = 180$ by $4$ .

$\frac{(3 x + 15) \cdot (4 x)}{4} = \frac{180}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{(3 x + 15) \cdot (4 x)}{4} = \frac{180}{4}$

Step 2.2.1.2

Divide $(3 x + 15) \cdot (x)$ by $1$ .

$(3 x + 15) \cdot (x) = \frac{180}{4}$

Step 2.2.2

Apply the distributive property.

$3 x \cdot x + 15 x = \frac{180}{4}$

Step 2.2.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.3.1

Move $x$ .

$3 (x \cdot x) + 15 x = \frac{180}{4}$

Step 2.2.3.2

Multiply $x$ by $x$ .

$3 x^{2} + 15 x = \frac{180}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $180$ by $4$ .

$3 x^{2} + 15 x = 45$

Step 3

Subtract $45$ from both sides of the equation.

$3 x^{2} + 15 x - 45 = 0$

Step 4

Factor $3$ out of $3 x^{2} + 15 x - 45$ .

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Step 4.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 15 x - 45 = 0$

Step 4.2

Factor $3$ out of $15 x$ .

$3 (x^{2}) + 3 (5 x) - 45 = 0$

Step 4.3

Factor $3$ out of $- 45$ .

$3 x^{2} + 3 (5 x) + 3 \cdot - 15 = 0$

Step 4.4

Factor $3$ out of $3 x^{2} + 3 (5 x)$ .

$3 (x^{2} + 5 x) + 3 \cdot - 15 = 0$

Step 4.5

Factor $3$ out of $3 (x^{2} + 5 x) + 3 \cdot - 15$ .

$3 (x^{2} + 5 x - 15) = 0$

Step 5

Divide each term in $3 (x^{2} + 5 x - 15) = 0$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 (x^{2} + 5 x - 15) = 0$ by $3$ .

$\frac{3 (x^{2} + 5 x - 15)}{3} = \frac{0}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} + 5 x - 15)}{3} = \frac{0}{3}$

Step 5.2.1.2

Divide $x^{2} + 5 x - 15$ by $1$ .

$x^{2} + 5 x - 15 = \frac{0}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $3$ .

$x^{2} + 5 x - 15 = 0$

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 $a = 1$ , $b = 5$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 15)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 15$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 15}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 15$ .

$x = \frac{- 5 \pm \sqrt{25 + 60}}{2 \cdot 1}$

Step 8.1.3

Add $25$ and $60$ .

$x = \frac{- 5 \pm \sqrt{85}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{85}}{2}$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 15$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 15}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 15$ .

$x = \frac{- 5 \pm \sqrt{25 + 60}}{2 \cdot 1}$

Step 9.1.3

Add $25$ and $60$ .

$x = \frac{- 5 \pm \sqrt{85}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{85}}{2}$

Step 9.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{85}}{2}$

Step 9.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{85}}{2}$

Step 9.5

Factor $- 1$ out of $\sqrt{85}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{85})}{2}$

Step 9.6

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{85})$ .

$x = \frac{- 1 (5 - \sqrt{85})}{2}$

Step 9.7

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{85}}{2}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot - 15$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 15}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $- 15$ .

$x = \frac{- 5 \pm \sqrt{25 + 60}}{2 \cdot 1}$

Step 10.1.3

Add $25$ and $60$ .

$x = \frac{- 5 \pm \sqrt{85}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{85}}{2}$

Step 10.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{85}}{2}$

Step 10.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{85}}{2}$

Step 10.5

Factor $- 1$ out of $- \sqrt{85}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{85})}{2}$

Step 10.6

Factor $- 1$ out of $- 1 (5) - (\sqrt{85})$ .

$x = \frac{- 1 (5 + \sqrt{85})}{2}$

Step 10.7

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{85}}{2}$

Step 11

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{85}}{2}, - \frac{5 + \sqrt{85}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 - \sqrt{85}}{2}, - \frac{5 + \sqrt{85}}{2}$

Decimal Form:

$x = 2.10977222 \dots, - 7.10977222 \dots$