Solve by Completing the Square 9x^2-24x=-13

$9 x^{2} - 24 x = - 13$

Step 1

Divide each term in $9 x^{2} - 24 x = - 13$ by $9$ and simplify.

Divide each term in $9 x^{2} - 24 x = - 13$ by $9$ .

$\frac{9 x^{2}}{9} + \frac{- 24 x}{9} = \frac{- 13}{9}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x^{2}}{9} + \frac{- 24 x}{9} = \frac{- 13}{9}$

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

$x^{2} + \frac{- 24 x}{9} = \frac{- 13}{9}$

Cancel the common factor of $- 24$ and $9$ .

Factor $3$ out of $- 24 x$ .

$x^{2} + \frac{3 (- 8 x)}{9} = \frac{- 13}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$x^{2} + \frac{3 (- 8 x)}{3 (3)} = \frac{- 13}{9}$

Cancel the common factor.

$x^{2} + \frac{3 (- 8 x)}{3 \cdot 3} = \frac{- 13}{9}$

Rewrite the expression.

$x^{2} + \frac{- 8 x}{3} = \frac{- 13}{9}$

Move the negative in front of the fraction.

$x^{2} - \frac{8 x}{3} = \frac{- 13}{9}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} - \frac{8 x}{3} = - \frac{13}{9}$

Step 2

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

${(\frac{b}{2})}^{2} = {(- \frac{4}{3})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - \frac{8 x}{3} + {(- \frac{4}{3})}^{2} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

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

Apply the product rule to $- \frac{4}{3}$ .

$x^{2} - \frac{8 x}{3} + {(- 1)}^{2} {(\frac{4}{3})}^{2} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

Apply the product rule to $\frac{4}{3}$ .

$x^{2} - \frac{8 x}{3} + {(- 1)}^{2} \frac{4^{2}}{3^{2}} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

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

$x^{2} - \frac{8 x}{3} + 1 \frac{4^{2}}{3^{2}} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

Multiply $\frac{4^{2}}{3^{2}}$ by $1$ .

$x^{2} - \frac{8 x}{3} + \frac{4^{2}}{3^{2}} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

Raise $4$ to the power of $2$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{3^{2}} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

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

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + {(- \frac{4}{3})}^{2}$

Simplify the right side.

Simplify $- \frac{13}{9} + {(- \frac{4}{3})}^{2}$ .

Simplify each term.

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

Apply the product rule to $- \frac{4}{3}$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + {(- 1)}^{2} {(\frac{4}{3})}^{2}$

Apply the product rule to $\frac{4}{3}$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + {(- 1)}^{2} \frac{4^{2}}{3^{2}}$

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

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + 1 \frac{4^{2}}{3^{2}}$

Multiply $\frac{4^{2}}{3^{2}}$ by $1$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + \frac{4^{2}}{3^{2}}$

Raise $4$ to the power of $2$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + \frac{16}{3^{2}}$

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

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = - \frac{13}{9} + \frac{16}{9}$

Simplify terms.

Combine the numerators over the common denominator.

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = \frac{- 13 + 16}{9}$

Add $- 13$ and $16$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = \frac{3}{9}$

Cancel the common factor of $3$ and $9$ .

Factor $3$ out of $3$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = \frac{3 (1)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = \frac{3 \cdot 1}{3 \cdot 3}$

Cancel the common factor.

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = \frac{3 \cdot 1}{3 \cdot 3}$

Rewrite the expression.

$x^{2} - \frac{8 x}{3} + \frac{16}{9} = \frac{1}{3}$

Step 5

Factor the perfect trinomial square into ${(x - \frac{4}{3})}^{2}$ .

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

Step 6

Solve the equation for $x$ .

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

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

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

$x - \frac{4}{3} = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x - \frac{4}{3} = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x - \frac{4}{3} = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - \frac{4}{3} = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

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

Evaluate the exponent.

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

Add $\frac{4}{3}$ to both sides of the equation.

$x = \pm \frac{\sqrt{3}}{3} + \frac{4}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{3}}{3} + \frac{4}{3}$

Decimal Form:

$x = 1.91068360 \dots, 0.75598306 \dots$

Solve by Completing the Square 9x^2-24x=-13

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