Precalculus Examples

$3 x^{2} - 2 x - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Move the negative in front of the fraction.

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{1}{3})}^{2}$

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Simplify each term.

Tap for more steps...

Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 5.1.1.1.1

Apply the product rule to $- \frac{1}{3}$ .

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

Step 5.1.1.1.2

Apply the product rule to $\frac{1}{3}$ .

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

Step 5.1.1.2

Raise $- 1$ to the power of $2$ .

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

Step 5.1.1.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

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

Step 5.1.1.4

One to any power is one.

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

Step 5.1.1.5

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

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

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 5.2.1.1.1.1

Apply the product rule to $- \frac{1}{3}$ .

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

Step 5.2.1.1.1.2

Apply the product rule to $\frac{1}{3}$ .

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

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

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

Step 5.2.1.1.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

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

Step 5.2.1.1.4

One to any power is one.

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

Step 5.2.1.1.5

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

Tap for more steps...

Step 5.2.1.3.1

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

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

Step 5.2.1.3.2

Multiply $3$ by $3$ .

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

Step 5.2.1.4

Combine the numerators over the common denominator.

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

Step 5.2.1.5

Add $3$ and $1$ .

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

Step 6

Factor the perfect trinomial square into ${(x - \frac{1}{3})}^{2}$ .

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

Step 7

Solve the equation for $x$ .

Tap for more steps...

Step 7.1

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

$x - \frac{1}{3} = \pm \sqrt{\frac{4}{9}}$

Step 7.2

Simplify $\pm \sqrt{\frac{4}{9}}$ .

Tap for more steps...

Step 7.2.1

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$x - \frac{1}{3} = \pm \frac{\sqrt{4}}{\sqrt{9}}$

Step 7.2.2

Simplify the numerator.

Tap for more steps...

Step 7.2.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.2.2.2

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

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

Step 7.2.3

Simplify the denominator.

Tap for more steps...

Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

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

Tap for more steps...

Step 7.3.2.1

Add $\frac{1}{3}$ to both sides of the equation.

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

Step 7.3.2.2

Combine the numerators over the common denominator.

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

Step 7.3.2.3

Add $2$ and $1$ .

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

Step 7.3.2.4

Divide $3$ by $3$ .

$x = 1$

Step 7.3.3

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

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

Step 7.3.4

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

Tap for more steps...

Step 7.3.4.1

Add $\frac{1}{3}$ to both sides of the equation.

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

Step 7.3.4.2

Combine the numerators over the common denominator.

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

Step 7.3.4.3

Add $- 2$ and $1$ .

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

Step 7.3.4.4

Move the negative in front of the fraction.

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

Step 7.3.5

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

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