Pre-Algebra Examples

$\frac{3 x - 8}{5} - (\frac{2 x - 6}{8} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1

Simplify $\frac{3 x - 8}{5} - (\frac{2 x - 6}{8} - \frac{x \cdot x}{6})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Cancel the common factor of $2 x - 6$ and $8$ .

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Step 1.1.1.1.1

Factor $2$ out of $2 x$ .

$\frac{3 x - 8}{5} - (\frac{2 (x) - 6}{8} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.1.1.2

Factor $2$ out of $- 6$ .

$\frac{3 x - 8}{5} - (\frac{2 x + 2 \cdot - 3}{8} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.1.1.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$\frac{3 x - 8}{5} - (\frac{2 (x - 3)}{8} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.1.1.4

Cancel the common factors.

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Step 1.1.1.1.4.1

Factor $2$ out of $8$ .

$\frac{3 x - 8}{5} - (\frac{2 (x - 3)}{2 (4)} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.1.1.4.2

Cancel the common factor.

$\frac{3 x - 8}{5} - (\frac{2 (x - 3)}{2 \cdot 4} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.1.1.4.3

Rewrite the expression.

$\frac{3 x - 8}{5} - (\frac{x - 3}{4} - \frac{x \cdot x}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.1.2

Multiply $x$ by $x$ .

$\frac{3 x - 8}{5} - (\frac{x - 3}{4} - \frac{x^{2}}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.2

To write $\frac{x - 3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x - 8}{5} - (\frac{x - 3}{4} \cdot \frac{3}{3} - \frac{x^{2}}{6}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.3

To write $- \frac{x^{2}}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x - 8}{5} - (\frac{x - 3}{4} \cdot \frac{3}{3} - \frac{x^{2}}{6} \cdot \frac{2}{2}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.4.1

Multiply $\frac{x - 3}{4}$ by $\frac{3}{3}$ .

$\frac{3 x - 8}{5} - (\frac{(x - 3) \cdot 3}{4 \cdot 3} - \frac{x^{2}}{6} \cdot \frac{2}{2}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.4.2

Multiply $4$ by $3$ .

$\frac{3 x - 8}{5} - (\frac{(x - 3) \cdot 3}{12} - \frac{x^{2}}{6} \cdot \frac{2}{2}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.4.3

Multiply $\frac{x^{2}}{6}$ by $\frac{2}{2}$ .

$\frac{3 x - 8}{5} - (\frac{(x - 3) \cdot 3}{12} - \frac{x^{2} \cdot 2}{6 \cdot 2}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.4.4

Multiply $6$ by $2$ .

$\frac{3 x - 8}{5} - (\frac{(x - 3) \cdot 3}{12} - \frac{x^{2} \cdot 2}{12}) = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{3 x - 8}{5} - \frac{(x - 3) \cdot 3 - x^{2} \cdot 2}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Apply the distributive property.

$\frac{3 x - 8}{5} - \frac{x \cdot 3 - 3 \cdot 3 - x^{2} \cdot 2}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.6.2

Move $3$ to the left of $x$ .

$\frac{3 x - 8}{5} - \frac{3 \cdot x - 3 \cdot 3 - x^{2} \cdot 2}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.6.3

Multiply $- 3$ by $3$ .

$\frac{3 x - 8}{5} - \frac{3 \cdot x - 9 - x^{2} \cdot 2}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.6.4

Multiply $2$ by $- 1$ .

$\frac{3 x - 8}{5} - \frac{3 x - 9 - 2 x^{2}}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.1.6.5

Reorder terms.

$\frac{3 x - 8}{5} - \frac{- 2 x^{2} + 3 x - 9}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.2

To write $\frac{3 x - 8}{5}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{3 x - 8}{5} \cdot \frac{12}{12} - \frac{- 2 x^{2} + 3 x - 9}{12} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.3

To write $- \frac{- 2 x^{2} + 3 x - 9}{12}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 x - 8}{5} \cdot \frac{12}{12} - \frac{- 2 x^{2} + 3 x - 9}{12} \cdot \frac{5}{5} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.4

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{3 x - 8}{5}$ by $\frac{12}{12}$ .

$\frac{(3 x - 8) \cdot 12}{5 \cdot 12} - \frac{- 2 x^{2} + 3 x - 9}{12} \cdot \frac{5}{5} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.4.2

Multiply $5$ by $12$ .

$\frac{(3 x - 8) \cdot 12}{60} - \frac{- 2 x^{2} + 3 x - 9}{12} \cdot \frac{5}{5} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.4.3

Multiply $\frac{- 2 x^{2} + 3 x - 9}{12}$ by $\frac{5}{5}$ .

$\frac{(3 x - 8) \cdot 12}{60} - \frac{(- 2 x^{2} + 3 x - 9) \cdot 5}{12 \cdot 5} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.4.4

Multiply $12$ by $5$ .

$\frac{(3 x - 8) \cdot 12}{60} - \frac{(- 2 x^{2} + 3 x - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{(3 x - 8) \cdot 12 - (- 2 x^{2} + 3 x - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

$\frac{3 x \cdot 12 - 8 \cdot 12 - (- 2 x^{2} + 3 x - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.2

Multiply $12$ by $3$ .

$\frac{36 x - 8 \cdot 12 - (- 2 x^{2} + 3 x - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.3

Multiply $- 8$ by $12$ .

$\frac{36 x - 96 - (- 2 x^{2} + 3 x - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.4

Apply the distributive property.

$\frac{36 x - 96 + (- (- 2 x^{2}) - (3 x) - - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.5

Simplify.

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Step 1.6.5.1

Multiply $- 2$ by $- 1$ .

$\frac{36 x - 96 + (2 x^{2} - (3 x) - - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.5.2

Multiply $3$ by $- 1$ .

$\frac{36 x - 96 + (2 x^{2} - 3 x - - 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.5.3

Multiply $- 1$ by $- 9$ .

$\frac{36 x - 96 + (2 x^{2} - 3 x + 9) \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.6

Apply the distributive property.

$\frac{36 x - 96 + 2 x^{2} \cdot 5 - 3 x \cdot 5 + 9 \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.7

Simplify.

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Step 1.6.7.1

Multiply $5$ by $2$ .

$\frac{36 x - 96 + 10 x^{2} - 3 x \cdot 5 + 9 \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.7.2

Multiply $5$ by $- 3$ .

$\frac{36 x - 96 + 10 x^{2} - 15 x + 9 \cdot 5}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.7.3

Multiply $9$ by $5$ .

$\frac{36 x - 96 + 10 x^{2} - 15 x + 45}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.8

Subtract $15 x$ from $36 x$ .

$\frac{21 x - 96 + 10 x^{2} + 45}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.9

Add $- 96$ and $45$ .

$\frac{21 x + 10 x^{2} - 51}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 1.6.10

Reorder terms.

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{3 x + 4}{15} + \frac{x - 3}{4}$

Step 2

Simplify $\frac{3 x + 4}{15} + \frac{x - 3}{4}$ .

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Step 2.1

To write $\frac{3 x + 4}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{3 x + 4}{15} \cdot \frac{4}{4} + \frac{x - 3}{4}$

Step 2.2

To write $\frac{x - 3}{4}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{3 x + 4}{15} \cdot \frac{4}{4} + \frac{x - 3}{4} \cdot \frac{15}{15}$

Step 2.3

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{3 x + 4}{15}$ by $\frac{4}{4}$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{(3 x + 4) \cdot 4}{15 \cdot 4} + \frac{x - 3}{4} \cdot \frac{15}{15}$

Step 2.3.2

Multiply $15$ by $4$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{(3 x + 4) \cdot 4}{60} + \frac{x - 3}{4} \cdot \frac{15}{15}$

Step 2.3.3

Multiply $\frac{x - 3}{4}$ by $\frac{15}{15}$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{(3 x + 4) \cdot 4}{60} + \frac{(x - 3) \cdot 15}{4 \cdot 15}$

Step 2.3.4

Multiply $4$ by $15$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{(3 x + 4) \cdot 4}{60} + \frac{(x - 3) \cdot 15}{60}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{(3 x + 4) \cdot 4 + (x - 3) \cdot 15}{60}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{3 x \cdot 4 + 4 \cdot 4 + (x - 3) \cdot 15}{60}$

Step 2.5.2

Multiply $4$ by $3$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{12 x + 4 \cdot 4 + (x - 3) \cdot 15}{60}$

Step 2.5.3

Multiply $4$ by $4$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{12 x + 16 + (x - 3) \cdot 15}{60}$

Step 2.5.4

Apply the distributive property.

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{12 x + 16 + x \cdot 15 - 3 \cdot 15}{60}$

Step 2.5.5

Move $15$ to the left of $x$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{12 x + 16 + 15 \cdot x - 3 \cdot 15}{60}$

Step 2.5.6

Multiply $- 3$ by $15$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{12 x + 16 + 15 \cdot x - 45}{60}$

Step 2.5.7

Add $12 x$ and $15 x$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{27 x + 16 - 45}{60}$

Step 2.5.8

Subtract $45$ from $16$ .

$\frac{10 x^{2} + 21 x - 51}{60} = \frac{27 x - 29}{60}$

Step 3

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

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Step 3.1

Subtract $\frac{27 x - 29}{60}$ from both sides of the equation.

$\frac{10 x^{2} + 21 x - 51}{60} - \frac{27 x - 29}{60} = 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{10 x^{2} + 21 x - 51 - (27 x - 29)}{60} = 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{10 x^{2} + 21 x - 51 - (27 x) - - 29}{60} = 0$

Step 3.3.2

Multiply $27$ by $- 1$ .

$\frac{10 x^{2} + 21 x - 51 - 27 x - - 29}{60} = 0$

Step 3.3.3

Multiply $- 1$ by $- 29$ .

$\frac{10 x^{2} + 21 x - 51 - 27 x + 29}{60} = 0$

Step 3.4

Subtract $27 x$ from $21 x$ .

$\frac{10 x^{2} - 6 x - 51 + 29}{60} = 0$

Step 3.5

Add $- 51$ and $29$ .

$\frac{10 x^{2} - 6 x - 22}{60} = 0$

Step 3.6

Cancel the common factor of $10 x^{2} - 6 x - 22$ and $60$ .

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Step 3.6.1

Factor $2$ out of $10 x^{2}$ .

$\frac{2 (5 x^{2}) - 6 x - 22}{60} = 0$

Step 3.6.2

Factor $2$ out of $- 6 x$ .

$\frac{2 (5 x^{2}) + 2 (- 3 x) - 22}{60} = 0$

Step 3.6.3

Factor $2$ out of $2 (5 x^{2}) + 2 (- 3 x)$ .

$\frac{2 (5 x^{2} - 3 x) - 22}{60} = 0$

Step 3.6.4

Factor $2$ out of $- 22$ .

$\frac{2 (5 x^{2} - 3 x) + 2 (- 11)}{60} = 0$

Step 3.6.5

Factor $2$ out of $2 (5 x^{2} - 3 x) + 2 (- 11)$ .

$\frac{2 (5 x^{2} - 3 x - 11)}{60} = 0$

Step 3.6.6

Cancel the common factors.

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Step 3.6.6.1

Factor $2$ out of $60$ .

$\frac{2 (5 x^{2} - 3 x - 11)}{2 (30)} = 0$

Step 3.6.6.2

Cancel the common factor.

$\frac{2 (5 x^{2} - 3 x - 11)}{2 \cdot 30} = 0$

Step 3.6.6.3

Rewrite the expression.

$\frac{5 x^{2} - 3 x - 11}{30} = 0$

Step 4

Set the numerator equal to zero.

$5 x^{2} - 3 x - 11 = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Use the quadratic formula to find the solutions.

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

Step 5.2

Substitute the values $a = 5$ , $b = - 3$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (5 \cdot - 11)}}{2 \cdot 5}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the numerator.

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Step 5.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 5 \cdot - 11}}{2 \cdot 5}$

Step 5.3.1.2

Multiply $- 4 \cdot 5 \cdot - 11$ .

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Step 5.3.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 20 \cdot - 11}}{2 \cdot 5}$

Step 5.3.1.2.2

Multiply $- 20$ by $- 11$ .

$x = \frac{3 \pm \sqrt{9 + 220}}{2 \cdot 5}$

Step 5.3.1.3

Add $9$ and $220$ .

$x = \frac{3 \pm \sqrt{229}}{2 \cdot 5}$

Step 5.3.2

Multiply $2$ by $5$ .

$x = \frac{3 \pm \sqrt{229}}{10}$

Step 5.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.4.1

Simplify the numerator.

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Step 5.4.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 5 \cdot - 11}}{2 \cdot 5}$

Step 5.4.1.2

Multiply $- 4 \cdot 5 \cdot - 11$ .

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Step 5.4.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 20 \cdot - 11}}{2 \cdot 5}$

Step 5.4.1.2.2

Multiply $- 20$ by $- 11$ .

$x = \frac{3 \pm \sqrt{9 + 220}}{2 \cdot 5}$

Step 5.4.1.3

Add $9$ and $220$ .

$x = \frac{3 \pm \sqrt{229}}{2 \cdot 5}$

Step 5.4.2

Multiply $2$ by $5$ .

$x = \frac{3 \pm \sqrt{229}}{10}$

Step 5.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{229}}{10}$

Step 5.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 5 \cdot - 11}}{2 \cdot 5}$

Step 5.5.1.2

Multiply $- 4 \cdot 5 \cdot - 11$ .

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Step 5.5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 20 \cdot - 11}}{2 \cdot 5}$

Step 5.5.1.2.2

Multiply $- 20$ by $- 11$ .

$x = \frac{3 \pm \sqrt{9 + 220}}{2 \cdot 5}$

Step 5.5.1.3

Add $9$ and $220$ .

$x = \frac{3 \pm \sqrt{229}}{2 \cdot 5}$

Step 5.5.2

Multiply $2$ by $5$ .

$x = \frac{3 \pm \sqrt{229}}{10}$

Step 5.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{229}}{10}$

Step 5.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{229}}{10}, \frac{3 - \sqrt{229}}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{229}}{10}, \frac{3 - \sqrt{229}}{10}$

Decimal Form:

$x = 1.81327459 \dots, - 1.21327459 \dots$