Pre-Algebra Examples

$6 x^{2} + 60 x + 900 = 902$

Step 1

Subtract $902$ from both sides of the equation.

$6 x^{2} + 60 x + 900 - 902 = 0$

Step 2

Subtract $902$ from $900$ .

$6 x^{2} + 60 x - 2 = 0$

Step 3

Factor $2$ out of $6 x^{2} + 60 x - 2$ .

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Step 3.1

Factor $2$ out of $6 x^{2}$ .

$2 (3 x^{2}) + 60 x - 2 = 0$

Step 3.2

Factor $2$ out of $60 x$ .

$2 (3 x^{2}) + 2 (30 x) - 2 = 0$

Step 3.3

Factor $2$ out of $- 2$ .

$2 (3 x^{2}) + 2 (30 x) + 2 (- 1) = 0$

Step 3.4

Factor $2$ out of $2 (3 x^{2}) + 2 (30 x)$ .

$2 (3 x^{2} + 30 x) + 2 (- 1) = 0$

Step 3.5

Factor $2$ out of $2 (3 x^{2} + 30 x) + 2 (- 1)$ .

$2 (3 x^{2} + 30 x - 1) = 0$

Step 4

Divide each term in $2 (3 x^{2} + 30 x - 1) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (3 x^{2} + 30 x - 1) = 0$ by $2$ .

$\frac{2 (3 x^{2} + 30 x - 1)}{2} = \frac{0}{2}$

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.

$\frac{2 (3 x^{2} + 30 x - 1)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $3 x^{2} + 30 x - 1$ by $1$ .

$3 x^{2} + 30 x - 1 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$3 x^{2} + 30 x - 1 = 0$

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 $a = 3$ , $b = 30$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 30 \pm \sqrt{30^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 30 \pm \sqrt{900 - 12 \cdot - 1}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $- 1$ .

$x = \frac{- 30 \pm \sqrt{900 + 12}}{2 \cdot 3}$

Step 7.1.3

Add $900$ and $12$ .

$x = \frac{- 30 \pm \sqrt{912}}{2 \cdot 3}$

Step 7.1.4

Rewrite $912$ as $4^{2} \cdot 57$ .

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Step 7.1.4.1

Factor $16$ out of $912$ .

$x = \frac{- 30 \pm \sqrt{16 (57)}}{2 \cdot 3}$

Step 7.1.4.2

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

$x = \frac{- 30 \pm \sqrt{4^{2} \cdot 57}}{2 \cdot 3}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 30 \pm 4 \sqrt{57}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$x = \frac{- 30 \pm 4 \sqrt{57}}{6}$

Step 7.3

Simplify $\frac{- 30 \pm 4 \sqrt{57}}{6}$ .

$x = \frac{- 15 \pm 2 \sqrt{57}}{3}$

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 $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 8.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 30 \pm \sqrt{900 - 12 \cdot - 1}}{2 \cdot 3}$

Step 8.1.2.2

Multiply $- 12$ by $- 1$ .

$x = \frac{- 30 \pm \sqrt{900 + 12}}{2 \cdot 3}$

Step 8.1.3

Add $900$ and $12$ .

$x = \frac{- 30 \pm \sqrt{912}}{2 \cdot 3}$

Step 8.1.4

Rewrite $912$ as $4^{2} \cdot 57$ .

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Step 8.1.4.1

Factor $16$ out of $912$ .

$x = \frac{- 30 \pm \sqrt{16 (57)}}{2 \cdot 3}$

Step 8.1.4.2

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

$x = \frac{- 30 \pm \sqrt{4^{2} \cdot 57}}{2 \cdot 3}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 30 \pm 4 \sqrt{57}}{2 \cdot 3}$

Step 8.2

Multiply $2$ by $3$ .

$x = \frac{- 30 \pm 4 \sqrt{57}}{6}$

Step 8.3

Simplify $\frac{- 30 \pm 4 \sqrt{57}}{6}$ .

$x = \frac{- 15 \pm 2 \sqrt{57}}{3}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{- 15 + 2 \sqrt{57}}{3}$

Step 8.5

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 1 \cdot 15 + 2 \sqrt{57}}{3}$

Step 8.6

Factor $- 1$ out of $2 \sqrt{57}$ .

$x = \frac{- 1 \cdot 15 - (- 2 \sqrt{57})}{3}$

Step 8.7

Factor $- 1$ out of $- 1 (15) - (- 2 \sqrt{57})$ .

$x = \frac{- 1 (15 - 2 \sqrt{57})}{3}$

Step 8.8

Move the negative in front of the fraction.

$x = - \frac{15 - 2 \sqrt{57}}{3}$

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 $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 9.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 9.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 30 \pm \sqrt{900 - 12 \cdot - 1}}{2 \cdot 3}$

Step 9.1.2.2

Multiply $- 12$ by $- 1$ .

$x = \frac{- 30 \pm \sqrt{900 + 12}}{2 \cdot 3}$

Step 9.1.3

Add $900$ and $12$ .

$x = \frac{- 30 \pm \sqrt{912}}{2 \cdot 3}$

Step 9.1.4

Rewrite $912$ as $4^{2} \cdot 57$ .

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Step 9.1.4.1

Factor $16$ out of $912$ .

$x = \frac{- 30 \pm \sqrt{16 (57)}}{2 \cdot 3}$

Step 9.1.4.2

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

$x = \frac{- 30 \pm \sqrt{4^{2} \cdot 57}}{2 \cdot 3}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{- 30 \pm 4 \sqrt{57}}{2 \cdot 3}$

Step 9.2

Multiply $2$ by $3$ .

$x = \frac{- 30 \pm 4 \sqrt{57}}{6}$

Step 9.3

Simplify $\frac{- 30 \pm 4 \sqrt{57}}{6}$ .

$x = \frac{- 15 \pm 2 \sqrt{57}}{3}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{- 15 - 2 \sqrt{57}}{3}$

Step 9.5

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 1 \cdot 15 - 2 \sqrt{57}}{3}$

Step 9.6

Factor $- 1$ out of $- 2 \sqrt{57}$ .

$x = \frac{- 1 \cdot 15 - (2 \sqrt{57})}{3}$

Step 9.7

Factor $- 1$ out of $- 1 (15) - (2 \sqrt{57})$ .

$x = \frac{- 1 (15 + 2 \sqrt{57})}{3}$

Step 9.8

Move the negative in front of the fraction.

$x = - \frac{15 + 2 \sqrt{57}}{3}$

Step 10

The final answer is the combination of both solutions.

$x = - \frac{15 - 2 \sqrt{57}}{3}, - \frac{15 + 2 \sqrt{57}}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{15 - 2 \sqrt{57}}{3}, - \frac{15 + 2 \sqrt{57}}{3}$

Decimal Form:

$x = 0.03322295 \dots, - 10.03322295 \dots$