Pre-Algebra Examples

$60 x^{2} + 157 x = 5 x - 30$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $5 x$ from both sides of the equation.

$60 x^{2} + 157 x - 5 x = - 30$

Step 1.2

Subtract $5 x$ from $157 x$ .

$60 x^{2} + 152 x = - 30$

Step 2

Add $30$ to both sides of the equation.

$60 x^{2} + 152 x + 30 = 0$

Step 3

Factor $2$ out of $60 x^{2} + 152 x + 30$ .

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Step 3.1

Factor $2$ out of $60 x^{2}$ .

$2 (30 x^{2}) + 152 x + 30 = 0$

Step 3.2

Factor $2$ out of $152 x$ .

$2 (30 x^{2}) + 2 (76 x) + 30 = 0$

Step 3.3

Factor $2$ out of $30$ .

$2 (30 x^{2}) + 2 (76 x) + 2 (15) = 0$

Step 3.4

Factor $2$ out of $2 (30 x^{2}) + 2 (76 x)$ .

$2 (30 x^{2} + 76 x) + 2 (15) = 0$

Step 3.5

Factor $2$ out of $2 (30 x^{2} + 76 x) + 2 (15)$ .

$2 (30 x^{2} + 76 x + 15) = 0$

Step 4

Divide each term in $2 (30 x^{2} + 76 x + 15) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (30 x^{2} + 76 x + 15) = 0$ by $2$ .

$\frac{2 (30 x^{2} + 76 x + 15)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (30 x^{2} + 76 x + 15)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $30 x^{2} + 76 x + 15$ by $1$ .

$30 x^{2} + 76 x + 15 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$30 x^{2} + 76 x + 15 = 0$

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 $a = 30$ , $b = 76$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{- 76 \pm \sqrt{76^{2} - 4 \cdot (30 \cdot 15)}}{2 \cdot 30}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $76$ to the power of $2$ .

$x = \frac{- 76 \pm \sqrt{5776 - 4 \cdot 30 \cdot 15}}{2 \cdot 30}$

Step 7.1.2

Multiply $- 4 \cdot 30 \cdot 15$ .

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Step 7.1.2.1

Multiply $- 4$ by $30$ .

$x = \frac{- 76 \pm \sqrt{5776 - 120 \cdot 15}}{2 \cdot 30}$

Step 7.1.2.2

Multiply $- 120$ by $15$ .

$x = \frac{- 76 \pm \sqrt{5776 - 1800}}{2 \cdot 30}$

Step 7.1.3

Subtract $1800$ from $5776$ .

$x = \frac{- 76 \pm \sqrt{3976}}{2 \cdot 30}$

Step 7.1.4

Rewrite $3976$ as $2^{2} \cdot 994$ .

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Step 7.1.4.1

Factor $4$ out of $3976$ .

$x = \frac{- 76 \pm \sqrt{4 (994)}}{2 \cdot 30}$

Step 7.1.4.2

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

$x = \frac{- 76 \pm \sqrt{2^{2} \cdot 994}}{2 \cdot 30}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 76 \pm 2 \sqrt{994}}{2 \cdot 30}$

Step 7.2

Multiply $2$ by $30$ .

$x = \frac{- 76 \pm 2 \sqrt{994}}{60}$

Step 7.3

Simplify $\frac{- 76 \pm 2 \sqrt{994}}{60}$ .

$x = \frac{- 38 \pm \sqrt{994}}{30}$

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 $76$ to the power of $2$ .

$x = \frac{- 76 \pm \sqrt{5776 - 4 \cdot 30 \cdot 15}}{2 \cdot 30}$

Step 8.1.2

Multiply $- 4 \cdot 30 \cdot 15$ .

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Step 8.1.2.1

Multiply $- 4$ by $30$ .

$x = \frac{- 76 \pm \sqrt{5776 - 120 \cdot 15}}{2 \cdot 30}$

Step 8.1.2.2

Multiply $- 120$ by $15$ .

$x = \frac{- 76 \pm \sqrt{5776 - 1800}}{2 \cdot 30}$

Step 8.1.3

Subtract $1800$ from $5776$ .

$x = \frac{- 76 \pm \sqrt{3976}}{2 \cdot 30}$

Step 8.1.4

Rewrite $3976$ as $2^{2} \cdot 994$ .

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Step 8.1.4.1

Factor $4$ out of $3976$ .

$x = \frac{- 76 \pm \sqrt{4 (994)}}{2 \cdot 30}$

Step 8.1.4.2

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

$x = \frac{- 76 \pm \sqrt{2^{2} \cdot 994}}{2 \cdot 30}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 76 \pm 2 \sqrt{994}}{2 \cdot 30}$

Step 8.2

Multiply $2$ by $30$ .

$x = \frac{- 76 \pm 2 \sqrt{994}}{60}$

Step 8.3

Simplify $\frac{- 76 \pm 2 \sqrt{994}}{60}$ .

$x = \frac{- 38 \pm \sqrt{994}}{30}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{- 38 + \sqrt{994}}{30}$

Step 8.5

Rewrite $- 38$ as $- 1 (38)$ .

$x = \frac{- 1 \cdot 38 + \sqrt{994}}{30}$

Step 8.6

Factor $- 1$ out of $\sqrt{994}$ .

$x = \frac{- 1 \cdot 38 - 1 (- \sqrt{994})}{30}$

Step 8.7

Factor $- 1$ out of $- 1 (38) - 1 (- \sqrt{994})$ .

$x = \frac{- 1 (38 - \sqrt{994})}{30}$

Step 8.8

Move the negative in front of the fraction.

$x = - \frac{38 - \sqrt{994}}{30}$

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 $76$ to the power of $2$ .

$x = \frac{- 76 \pm \sqrt{5776 - 4 \cdot 30 \cdot 15}}{2 \cdot 30}$

Step 9.1.2

Multiply $- 4 \cdot 30 \cdot 15$ .

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Step 9.1.2.1

Multiply $- 4$ by $30$ .

$x = \frac{- 76 \pm \sqrt{5776 - 120 \cdot 15}}{2 \cdot 30}$

Step 9.1.2.2

Multiply $- 120$ by $15$ .

$x = \frac{- 76 \pm \sqrt{5776 - 1800}}{2 \cdot 30}$

Step 9.1.3

Subtract $1800$ from $5776$ .

$x = \frac{- 76 \pm \sqrt{3976}}{2 \cdot 30}$

Step 9.1.4

Rewrite $3976$ as $2^{2} \cdot 994$ .

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Step 9.1.4.1

Factor $4$ out of $3976$ .

$x = \frac{- 76 \pm \sqrt{4 (994)}}{2 \cdot 30}$

Step 9.1.4.2

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

$x = \frac{- 76 \pm \sqrt{2^{2} \cdot 994}}{2 \cdot 30}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{- 76 \pm 2 \sqrt{994}}{2 \cdot 30}$

Step 9.2

Multiply $2$ by $30$ .

$x = \frac{- 76 \pm 2 \sqrt{994}}{60}$

Step 9.3

Simplify $\frac{- 76 \pm 2 \sqrt{994}}{60}$ .

$x = \frac{- 38 \pm \sqrt{994}}{30}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{- 38 - \sqrt{994}}{30}$

Step 9.5

Rewrite $- 38$ as $- 1 (38)$ .

$x = \frac{- 1 \cdot 38 - \sqrt{994}}{30}$

Step 9.6

Factor $- 1$ out of $- \sqrt{994}$ .

$x = \frac{- 1 \cdot 38 - (\sqrt{994})}{30}$

Step 9.7

Factor $- 1$ out of $- 1 (38) - (\sqrt{994})$ .

$x = \frac{- 1 (38 + \sqrt{994})}{30}$

Step 9.8

Move the negative in front of the fraction.

$x = - \frac{38 + \sqrt{994}}{30}$

Step 10

The final answer is the combination of both solutions.

$x = - \frac{38 - \sqrt{994}}{30}, - \frac{38 + \sqrt{994}}{30}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{38 - \sqrt{994}}{30}, - \frac{38 + \sqrt{994}}{30}$

Decimal Form:

$x = - 0.21574114 \dots, - 2.31759218 \dots$