Pre-Algebra Examples

$x^{2} + \frac{6}{17} x + \frac{9}{289} = \frac{16}{289}$

Step 1

Combine $\frac{6}{17}$ and $x$ .

$x^{2} + \frac{6 x}{17} + \frac{9}{289} = \frac{16}{289}$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $\frac{16}{289}$ from both sides of the equation.

$x^{2} + \frac{6 x}{17} + \frac{9}{289} - \frac{16}{289} = 0$

Step 2.2

Simplify $x^{2} + \frac{6 x}{17} + \frac{9}{289} - \frac{16}{289}$ .

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Step 2.2.1

Combine the numerators over the common denominator.

$x^{2} + \frac{6 x}{17} + \frac{9 - 16}{289} = 0$

Step 2.2.2

Subtract $16$ from $9$ .

$x^{2} + \frac{6 x}{17} + \frac{- 7}{289} = 0$

Step 2.2.3

Move the negative in front of the fraction.

$x^{2} + \frac{6 x}{17} - \frac{7}{289} = 0$

Step 3

Multiply through by the least common denominator $289$ , then simplify.

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Step 3.1

Apply the distributive property.

$289 x^{2} + 289 (\frac{6 x}{17}) + 289 (- \frac{7}{289}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $17$ .

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Step 3.2.1.1

Factor $17$ out of $289$ .

$289 x^{2} + 17 (17) (\frac{6 x}{17}) + 289 (- \frac{7}{289}) = 0$

Step 3.2.1.2

Cancel the common factor.

$289 x^{2} + 17 \cdot (17 (\frac{6 x}{17})) + 289 (- \frac{7}{289}) = 0$

Step 3.2.1.3

Rewrite the expression.

$289 x^{2} + 17 (6 x) + 289 (- \frac{7}{289}) = 0$

Step 3.2.2

Multiply $6$ by $17$ .

$289 x^{2} + 102 x + 289 (- \frac{7}{289}) = 0$

Step 3.2.3

Cancel the common factor of $289$ .

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Step 3.2.3.1

Move the leading negative in $- \frac{7}{289}$ into the numerator.

$289 x^{2} + 102 x + 289 (\frac{- 7}{289}) = 0$

Step 3.2.3.2

Cancel the common factor.

$289 x^{2} + 102 x + 289 (\frac{- 7}{289}) = 0$

Step 3.2.3.3

Rewrite the expression.

$289 x^{2} + 102 x - 7 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 289$ , $b = 102$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 102 \pm \sqrt{102^{2} - 4 \cdot (289 \cdot - 7)}}{2 \cdot 289}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $102$ to the power of $2$ .

$x = \frac{- 102 \pm \sqrt{10404 - 4 \cdot 289 \cdot - 7}}{2 \cdot 289}$

Step 6.1.2

Multiply $- 4 \cdot 289 \cdot - 7$ .

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Step 6.1.2.1

Multiply $- 4$ by $289$ .

$x = \frac{- 102 \pm \sqrt{10404 - 1156 \cdot - 7}}{2 \cdot 289}$

Step 6.1.2.2

Multiply $- 1156$ by $- 7$ .

$x = \frac{- 102 \pm \sqrt{10404 + 8092}}{2 \cdot 289}$

Step 6.1.3

Add $10404$ and $8092$ .

$x = \frac{- 102 \pm \sqrt{18496}}{2 \cdot 289}$

Step 6.1.4

Rewrite $18496$ as $136^{2}$ .

$x = \frac{- 102 \pm \sqrt{136^{2}}}{2 \cdot 289}$

Step 6.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 102 \pm 136}{2 \cdot 289}$

Step 6.2

Multiply $2$ by $289$ .

$x = \frac{- 102 \pm 136}{578}$

Step 6.3

Simplify $\frac{- 102 \pm 136}{578}$ .

$x = \frac{- 3 \pm 4}{17}$

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 $102$ to the power of $2$ .

$x = \frac{- 102 \pm \sqrt{10404 - 4 \cdot 289 \cdot - 7}}{2 \cdot 289}$

Step 7.1.2

Multiply $- 4 \cdot 289 \cdot - 7$ .

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Step 7.1.2.1

Multiply $- 4$ by $289$ .

$x = \frac{- 102 \pm \sqrt{10404 - 1156 \cdot - 7}}{2 \cdot 289}$

Step 7.1.2.2

Multiply $- 1156$ by $- 7$ .

$x = \frac{- 102 \pm \sqrt{10404 + 8092}}{2 \cdot 289}$

Step 7.1.3

Add $10404$ and $8092$ .

$x = \frac{- 102 \pm \sqrt{18496}}{2 \cdot 289}$

Step 7.1.4

Rewrite $18496$ as $136^{2}$ .

$x = \frac{- 102 \pm \sqrt{136^{2}}}{2 \cdot 289}$

Step 7.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 102 \pm 136}{2 \cdot 289}$

Step 7.2

Multiply $2$ by $289$ .

$x = \frac{- 102 \pm 136}{578}$

Step 7.3

Simplify $\frac{- 102 \pm 136}{578}$ .

$x = \frac{- 3 \pm 4}{17}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 4}{17}$

Step 7.5

Add $- 3$ and $4$ .

$x = \frac{1}{17}$

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 $102$ to the power of $2$ .

$x = \frac{- 102 \pm \sqrt{10404 - 4 \cdot 289 \cdot - 7}}{2 \cdot 289}$

Step 8.1.2

Multiply $- 4 \cdot 289 \cdot - 7$ .

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Step 8.1.2.1

Multiply $- 4$ by $289$ .

$x = \frac{- 102 \pm \sqrt{10404 - 1156 \cdot - 7}}{2 \cdot 289}$

Step 8.1.2.2

Multiply $- 1156$ by $- 7$ .

$x = \frac{- 102 \pm \sqrt{10404 + 8092}}{2 \cdot 289}$

Step 8.1.3

Add $10404$ and $8092$ .

$x = \frac{- 102 \pm \sqrt{18496}}{2 \cdot 289}$

Step 8.1.4

Rewrite $18496$ as $136^{2}$ .

$x = \frac{- 102 \pm \sqrt{136^{2}}}{2 \cdot 289}$

Step 8.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 102 \pm 136}{2 \cdot 289}$

Step 8.2

Multiply $2$ by $289$ .

$x = \frac{- 102 \pm 136}{578}$

Step 8.3

Simplify $\frac{- 102 \pm 136}{578}$ .

$x = \frac{- 3 \pm 4}{17}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 4}{17}$

Step 8.5

Subtract $4$ from $- 3$ .

$x = \frac{- 7}{17}$

Step 8.6

Move the negative in front of the fraction.

$x = - \frac{7}{17}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{1}{17}, - \frac{7}{17}$