Pre-Algebra Examples

$8 (x) = 4 x^{2} - 48 x + 7$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 48 x + 7 = 8 x$

Step 2

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

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Step 2.1

Subtract $8 x$ from both sides of the equation.

$4 x^{2} - 48 x + 7 - 8 x = 0$

Step 2.2

Subtract $8 x$ from $- 48 x$ .

$4 x^{2} - 56 x + 7 = 0$

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 $a = 4$ , $b = - 56$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{56 \pm \sqrt{{(- 56)}^{2} - 4 \cdot (4 \cdot 7)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 56$ to the power of $2$ .

$x = \frac{56 \pm \sqrt{3136 - 4 \cdot 4 \cdot 7}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot 7$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{56 \pm \sqrt{3136 - 16 \cdot 7}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $7$ .

$x = \frac{56 \pm \sqrt{3136 - 112}}{2 \cdot 4}$

Step 5.1.3

Subtract $112$ from $3136$ .

$x = \frac{56 \pm \sqrt{3024}}{2 \cdot 4}$

Step 5.1.4

Rewrite $3024$ as $12^{2} \cdot 21$ .

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Step 5.1.4.1

Factor $144$ out of $3024$ .

$x = \frac{56 \pm \sqrt{144 (21)}}{2 \cdot 4}$

Step 5.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{56 \pm \sqrt{12^{2} \cdot 21}}{2 \cdot 4}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{56 \pm 12 \sqrt{21}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{56 \pm 12 \sqrt{21}}{8}$

Step 5.3

Simplify $\frac{56 \pm 12 \sqrt{21}}{8}$ .

$x = \frac{14 \pm 3 \sqrt{21}}{2}$

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 $- 56$ to the power of $2$ .

$x = \frac{56 \pm \sqrt{3136 - 4 \cdot 4 \cdot 7}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot 7$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{56 \pm \sqrt{3136 - 16 \cdot 7}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $7$ .

$x = \frac{56 \pm \sqrt{3136 - 112}}{2 \cdot 4}$

Step 6.1.3

Subtract $112$ from $3136$ .

$x = \frac{56 \pm \sqrt{3024}}{2 \cdot 4}$

Step 6.1.4

Rewrite $3024$ as $12^{2} \cdot 21$ .

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Step 6.1.4.1

Factor $144$ out of $3024$ .

$x = \frac{56 \pm \sqrt{144 (21)}}{2 \cdot 4}$

Step 6.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{56 \pm \sqrt{12^{2} \cdot 21}}{2 \cdot 4}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{56 \pm 12 \sqrt{21}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{56 \pm 12 \sqrt{21}}{8}$

Step 6.3

Simplify $\frac{56 \pm 12 \sqrt{21}}{8}$ .

$x = \frac{14 \pm 3 \sqrt{21}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{14 + 3 \sqrt{21}}{2}$

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 $- 56$ to the power of $2$ .

$x = \frac{56 \pm \sqrt{3136 - 4 \cdot 4 \cdot 7}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot 7$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{56 \pm \sqrt{3136 - 16 \cdot 7}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $7$ .

$x = \frac{56 \pm \sqrt{3136 - 112}}{2 \cdot 4}$

Step 7.1.3

Subtract $112$ from $3136$ .

$x = \frac{56 \pm \sqrt{3024}}{2 \cdot 4}$

Step 7.1.4

Rewrite $3024$ as $12^{2} \cdot 21$ .

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Step 7.1.4.1

Factor $144$ out of $3024$ .

$x = \frac{56 \pm \sqrt{144 (21)}}{2 \cdot 4}$

Step 7.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{56 \pm \sqrt{12^{2} \cdot 21}}{2 \cdot 4}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{56 \pm 12 \sqrt{21}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{56 \pm 12 \sqrt{21}}{8}$

Step 7.3

Simplify $\frac{56 \pm 12 \sqrt{21}}{8}$ .

$x = \frac{14 \pm 3 \sqrt{21}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{14 - 3 \sqrt{21}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{14 + 3 \sqrt{21}}{2}, \frac{14 - 3 \sqrt{21}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{14 + 3 \sqrt{21}}{2}, \frac{14 - 3 \sqrt{21}}{2}$

Decimal Form:

$x = 13.87386354 \dots, 0.12613645 \dots$