Precalculus Examples

$3 x^{2} + 13 x + 10 = 10$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $10$ from both sides of the equation.

$3 x^{2} + 13 x = 10 - 10$

Step 1.2

Subtract $10$ from $10$ .

$3 x^{2} + 13 x = 0$

Step 2

Divide each term in $3 x^{2} + 13 x = 0$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} + 13 x = 0$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{13 x}{3} = \frac{0}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{13 x}{3} = \frac{0}{3}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{13 x}{3} = \frac{0}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $3$ .

$x^{2} + \frac{13 x}{3} = 0$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{13}{6})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{13 x}{3} + {(\frac{13}{6})}^{2} = 0 + {(\frac{13}{6})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $\frac{13}{6}$ .

$x^{2} + \frac{13 x}{3} + \frac{13^{2}}{6^{2}} = 0 + {(\frac{13}{6})}^{2}$

Step 5.1.1.2

Raise $13$ to the power of $2$ .

$x^{2} + \frac{13 x}{3} + \frac{169}{6^{2}} = 0 + {(\frac{13}{6})}^{2}$

Step 5.1.1.3

Raise $6$ to the power of $2$ .

$x^{2} + \frac{13 x}{3} + \frac{169}{36} = 0 + {(\frac{13}{6})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $0 + {(\frac{13}{6})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the product rule to $\frac{13}{6}$ .

$x^{2} + \frac{13 x}{3} + \frac{169}{36} = 0 + \frac{13^{2}}{6^{2}}$

Step 5.2.1.1.2

Raise $13$ to the power of $2$ .

$x^{2} + \frac{13 x}{3} + \frac{169}{36} = 0 + \frac{169}{6^{2}}$

Step 5.2.1.1.3

Raise $6$ to the power of $2$ .

$x^{2} + \frac{13 x}{3} + \frac{169}{36} = 0 + \frac{169}{36}$

Step 5.2.1.2

Add $0$ and $\frac{169}{36}$ .

$x^{2} + \frac{13 x}{3} + \frac{169}{36} = \frac{169}{36}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{13}{6})}^{2}$ .

${(x + \frac{13}{6})}^{2} = \frac{169}{36}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{13}{6} = \pm \sqrt{\frac{169}{36}}$

Step 7.2

Simplify $\pm \sqrt{\frac{169}{36}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{169}{36}}$ as $\frac{\sqrt{169}}{\sqrt{36}}$ .

$x + \frac{13}{6} = \pm \frac{\sqrt{169}}{\sqrt{36}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $169$ as $13^{2}$ .

$x + \frac{13}{6} = \pm \frac{\sqrt{13^{2}}}{\sqrt{36}}$

Step 7.2.2.2

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

$x + \frac{13}{6} = \pm \frac{13}{\sqrt{36}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $36$ as $6^{2}$ .

$x + \frac{13}{6} = \pm \frac{13}{\sqrt{6^{2}}}$

Step 7.2.3.2

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

$x + \frac{13}{6} = \pm \frac{13}{6}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x + \frac{13}{6} = \frac{13}{6}$

Step 7.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.2.1

Subtract $\frac{13}{6}$ from both sides of the equation.

$x = \frac{13}{6} - \frac{13}{6}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$x = \frac{13 - 13}{6}$

Step 7.3.2.3

Subtract $13$ from $13$ .

$x = \frac{0}{6}$

Step 7.3.2.4

Divide $0$ by $6$ .

$x = 0$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + \frac{13}{6} = - \frac{13}{6}$

Step 7.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.4.1

Subtract $\frac{13}{6}$ from both sides of the equation.

$x = - \frac{13}{6} - \frac{13}{6}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$x = \frac{- 13 - 13}{6}$

Step 7.3.4.3

Subtract $13$ from $- 13$ .

$x = \frac{- 26}{6}$

Step 7.3.4.4

Cancel the common factor of $- 26$ and $6$ .

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Step 7.3.4.4.1

Factor $2$ out of $- 26$ .

$x = \frac{2 (- 13)}{6}$

Step 7.3.4.4.2

Cancel the common factors.

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Step 7.3.4.4.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 13}{2 \cdot 3}$

Step 7.3.4.4.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 13}{2 \cdot 3}$

Step 7.3.4.4.2.3

Rewrite the expression.

$x = \frac{- 13}{3}$

Step 7.3.4.5

Move the negative in front of the fraction.

$x = - \frac{13}{3}$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 0, - \frac{13}{3}$