Pre-Algebra Examples

${(4 x + 2)}^{2} + {23.3}^{2} = x^{2}$

Step 1

Raise $23.3$ to the power of $2$ .

${(4 x + 2)}^{2} + 542.89 = x^{2}$

Step 2

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

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

${(4 x + 2)}^{2} + 542.89 - x^{2} = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(4 x + 2)}^{2}$ as $(4 x + 2) (4 x + 2)$ .

$(4 x + 2) (4 x + 2) + 542.89 - x^{2} = 0$

Step 2.2.2

Expand $(4 x + 2) (4 x + 2)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$4 x (4 x + 2) + 2 (4 x + 2) + 542.89 - x^{2} = 0$

Step 2.2.2.2

Apply the distributive property.

$4 x (4 x) + 4 x \cdot 2 + 2 (4 x + 2) + 542.89 - x^{2} = 0$

Step 2.2.2.3

Apply the distributive property.

$4 x (4 x) + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite using the commutative property of multiplication.

$4 \cdot 4 x \cdot x + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

Move $x$ .

$4 \cdot 4 (x \cdot x) + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3.1.2.2

Multiply $x$ by $x$ .

$4 \cdot 4 x^{2} + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3.1.3

Multiply $4$ by $4$ .

$16 x^{2} + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3.1.4

Multiply $2$ by $4$ .

$16 x^{2} + 8 x + 2 (4 x) + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3.1.5

Multiply $4$ by $2$ .

$16 x^{2} + 8 x + 8 x + 2 \cdot 2 + 542.89 - x^{2} = 0$

Step 2.2.3.1.6

Multiply $2$ by $2$ .

$16 x^{2} + 8 x + 8 x + 4 + 542.89 - x^{2} = 0$

Step 2.2.3.2

Add $8 x$ and $8 x$ .

$16 x^{2} + 16 x + 4 + 542.89 - x^{2} = 0$

Step 2.3

Subtract $x^{2}$ from $16 x^{2}$ .

$15 x^{2} + 16 x + 4 + 542.89 = 0$

Step 2.4

Add $4$ and $542.89$ .

$15 x^{2} + 16 x + 546.89 = 0$

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 $a = 15$ , $b = 16$ , and $c = 546.89$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (15 \cdot 546.89)}}{2 \cdot 15}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 15 \cdot 546.89}}{2 \cdot 15}$

Step 5.1.2

Multiply $- 4 \cdot 15 \cdot 546.89$ .

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Step 5.1.2.1

Multiply $- 4$ by $15$ .

$x = \frac{- 16 \pm \sqrt{256 - 60 \cdot 546.89}}{2 \cdot 15}$

Step 5.1.2.2

Multiply $- 60$ by $546.89$ .

$x = \frac{- 16 \pm \sqrt{256 - 32813.4}}{2 \cdot 15}$

Step 5.1.3

Subtract $32813.4$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 32557.4}}{2 \cdot 15}$

Step 5.1.4

Rewrite $- 32557.4$ as $- 1 (32557.4)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 32557.4}}{2 \cdot 15}$

Step 5.1.5

Rewrite $\sqrt{- 1 (32557.4)}$ as $\sqrt{- 1} \cdot \sqrt{32557.4}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{32557.4}}{2 \cdot 15}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 16 \pm i \sqrt{32557.4}}{2 \cdot 15}$

Step 5.2

Multiply $2$ by $15$ .

$x = \frac{- 16 \pm i \sqrt{32557.4}}{30}$

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 $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 15 \cdot 546.89}}{2 \cdot 15}$

Step 6.1.2

Multiply $- 4 \cdot 15 \cdot 546.89$ .

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Step 6.1.2.1

Multiply $- 4$ by $15$ .

$x = \frac{- 16 \pm \sqrt{256 - 60 \cdot 546.89}}{2 \cdot 15}$

Step 6.1.2.2

Multiply $- 60$ by $546.89$ .

$x = \frac{- 16 \pm \sqrt{256 - 32813.4}}{2 \cdot 15}$

Step 6.1.3

Subtract $32813.4$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 32557.4}}{2 \cdot 15}$

Step 6.1.4

Rewrite $- 32557.4$ as $- 1 (32557.4)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 32557.4}}{2 \cdot 15}$

Step 6.1.5

Rewrite $\sqrt{- 1 (32557.4)}$ as $\sqrt{- 1} \cdot \sqrt{32557.4}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{32557.4}}{2 \cdot 15}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 16 \pm i \sqrt{32557.4}}{2 \cdot 15}$

Step 6.2

Multiply $2$ by $15$ .

$x = \frac{- 16 \pm i \sqrt{32557.4}}{30}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 16 + i \sqrt{32557.4}}{30}$

Step 6.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 1 \cdot 16 + i \sqrt{32557.4}}{30}$

Step 6.5

Factor $- 1$ out of $i \sqrt{32557.4}$ .

$x = \frac{- 1 \cdot 16 - (- i \sqrt{32557.4})}{30}$

Step 6.6

Factor $- 1$ out of $- 1 (16) - (- i \sqrt{32557.4})$ .

$x = \frac{- 1 (16 - i \sqrt{32557.4})}{30}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{16 - i \sqrt{32557.4}}{30}$

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 $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 15 \cdot 546.89}}{2 \cdot 15}$

Step 7.1.2

Multiply $- 4 \cdot 15 \cdot 546.89$ .

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Step 7.1.2.1

Multiply $- 4$ by $15$ .

$x = \frac{- 16 \pm \sqrt{256 - 60 \cdot 546.89}}{2 \cdot 15}$

Step 7.1.2.2

Multiply $- 60$ by $546.89$ .

$x = \frac{- 16 \pm \sqrt{256 - 32813.4}}{2 \cdot 15}$

Step 7.1.3

Subtract $32813.4$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 32557.4}}{2 \cdot 15}$

Step 7.1.4

Rewrite $- 32557.4$ as $- 1 (32557.4)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 32557.4}}{2 \cdot 15}$

Step 7.1.5

Rewrite $\sqrt{- 1 (32557.4)}$ as $\sqrt{- 1} \cdot \sqrt{32557.4}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{32557.4}}{2 \cdot 15}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 16 \pm i \sqrt{32557.4}}{2 \cdot 15}$

Step 7.2

Multiply $2$ by $15$ .

$x = \frac{- 16 \pm i \sqrt{32557.4}}{30}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 16 - i \sqrt{32557.4}}{30}$

Step 7.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 1 \cdot 16 - i \sqrt{32557.4}}{30}$

Step 7.5

Factor $- 1$ out of $- i \sqrt{32557.4}$ .

$x = \frac{- 1 \cdot 16 - (i \sqrt{32557.4})}{30}$

Step 7.6

Factor $- 1$ out of $- 1 (16) - (i \sqrt{32557.4})$ .

$x = \frac{- 1 (16 + i \sqrt{32557.4})}{30}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{16 + i \sqrt{32557.4}}{30}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{16 - i \sqrt{32557.4}}{30}, - \frac{16 + i \sqrt{32557.4}}{30}$