Precalculus Examples

$3 x^{2} + 4 x + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$3 x^{2} + 4 x = - 1$

Step 2

Divide each term in $3 x^{2} + 4 x = - 1$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} + 4 x = - 1$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{4 x}{3} = \frac{- 1}{3}$

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.

$\frac{3 x^{2}}{3} + \frac{4 x}{3} = \frac{- 1}{3}$

Step 2.2.1.2

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

$x^{2} + \frac{4 x}{3} = \frac{- 1}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x^{2} + \frac{4 x}{3} = - \frac{1}{3}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{2}{3})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{4 x}{3} + {(\frac{2}{3})}^{2} = - \frac{1}{3} + {(\frac{2}{3})}^{2}$

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 $\frac{2}{3}$ .

$x^{2} + \frac{4 x}{3} + \frac{2^{2}}{3^{2}} = - \frac{1}{3} + {(\frac{2}{3})}^{2}$

Step 5.1.1.2

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

$x^{2} + \frac{4 x}{3} + \frac{4}{3^{2}} = - \frac{1}{3} + {(\frac{2}{3})}^{2}$

Step 5.1.1.3

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

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{1}{3} + {(\frac{2}{3})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{1}{3} + {(\frac{2}{3})}^{2}$ .

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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 $\frac{2}{3}$ .

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{1}{3} + \frac{2^{2}}{3^{2}}$

Step 5.2.1.1.2

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

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{1}{3} + \frac{4}{3^{2}}$

Step 5.2.1.1.3

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

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{1}{3} + \frac{4}{9}$

Step 5.2.1.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{1}{3} \cdot \frac{3}{3} + \frac{4}{9}$

Step 5.2.1.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{1}{3}$ by $\frac{3}{3}$ .

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{3}{3 \cdot 3} + \frac{4}{9}$

Step 5.2.1.3.2

Multiply $3$ by $3$ .

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = - \frac{3}{9} + \frac{4}{9}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = \frac{- 3 + 4}{9}$

Step 5.2.1.5

Add $- 3$ and $4$ .

$x^{2} + \frac{4 x}{3} + \frac{4}{9} = \frac{1}{9}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{2}{3})}^{2}$ .

${(x + \frac{2}{3})}^{2} = \frac{1}{9}$

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.

$x + \frac{2}{3} = \pm \sqrt{\frac{1}{9}}$

Step 7.2

Simplify $\pm \sqrt{\frac{1}{9}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$x + \frac{2}{3} = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Step 7.2.2

Any root of $1$ is $1$ .

$x + \frac{2}{3} = \pm \frac{1}{\sqrt{9}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $9$ as $3^{2}$ .

$x + \frac{2}{3} = \pm \frac{1}{\sqrt{3^{2}}}$

Step 7.2.3.2

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

$x + \frac{2}{3} = \pm \frac{1}{3}$

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.

$x + \frac{2}{3} = \frac{1}{3}$

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 $\frac{2}{3}$ from both sides of the equation.

$x = \frac{1}{3} - \frac{2}{3}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$x = \frac{1 - 2}{3}$

Step 7.3.2.3

Subtract $2$ from $1$ .

$x = \frac{- 1}{3}$

Step 7.3.2.4

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 7.3.3

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

$x + \frac{2}{3} = - \frac{1}{3}$

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 $\frac{2}{3}$ from both sides of the equation.

$x = - \frac{1}{3} - \frac{2}{3}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$x = \frac{- 1 - 2}{3}$

Step 7.3.4.3

Subtract $2$ from $- 1$ .

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

Step 7.3.4.4

Divide $- 3$ by $3$ .

$x = - 1$

Step 7.3.5

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

$x = - \frac{1}{3}, - 1$