Examples

$y = 2 x^{2} - 12 x + 9$

Step 1

Plug in $0$ for $y$ .

$0 = 2 x^{2} - 12 x + 9$

Step 2

Simplify the equation into a proper form to complete the square.

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Step 2.1

Remove parentheses.

$0 = 2 x^{2} - 12 x + 9$

Step 2.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} - 12 x + 9 = 0$

Step 2.3

Subtract $9$ from both sides of the equation.

$2 x^{2} - 12 x = - 9$

Step 3

Divide each term in $2 x^{2} - 12 x = - 9$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x^{2} - 12 x = - 9$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{- 12 x}{2} = \frac{- 9}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{- 12 x}{2} = \frac{- 9}{2}$

Step 3.2.1.1.2

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

$x^{2} + \frac{- 12 x}{2} = \frac{- 9}{2}$

Step 3.2.1.2

Cancel the common factor of $- 12$ and $2$ .

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Step 3.2.1.2.1

Factor $2$ out of $- 12 x$ .

$x^{2} + \frac{2 (- 6 x)}{2} = \frac{- 9}{2}$

Step 3.2.1.2.2

Cancel the common factors.

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Step 3.2.1.2.2.1

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (- 6 x)}{2 (1)} = \frac{- 9}{2}$

Step 3.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{2 (- 6 x)}{2 \cdot 1} = \frac{- 9}{2}$

Step 3.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{- 6 x}{1} = \frac{- 9}{2}$

Step 3.2.1.2.2.4

Divide $- 6 x$ by $1$ .

$x^{2} - 6 x = \frac{- 9}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$x^{2} - 6 x = - \frac{9}{2}$

Step 4

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

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

Step 5

Add the term to each side of the equation.

$x^{2} - 6 x + {(- 3)}^{2} = - \frac{9}{2} + {(- 3)}^{2}$

Step 6

Simplify the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

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

$x^{2} - 6 x + 9 = - \frac{9}{2} + {(- 3)}^{2}$

Step 6.2

Simplify the right side.

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Step 6.2.1

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

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Step 6.2.1.1

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

$x^{2} - 6 x + 9 = - \frac{9}{2} + 9$

Step 6.2.1.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} - 6 x + 9 = - \frac{9}{2} + 9 \cdot \frac{2}{2}$

Step 6.2.1.3

Combine $9$ and $\frac{2}{2}$ .

$x^{2} - 6 x + 9 = - \frac{9}{2} + \frac{9 \cdot 2}{2}$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 6 x + 9 = \frac{- 9 + 9 \cdot 2}{2}$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $9$ by $2$ .

$x^{2} - 6 x + 9 = \frac{- 9 + 18}{2}$

Step 6.2.1.5.2

Add $- 9$ and $18$ .

$x^{2} - 6 x + 9 = \frac{9}{2}$

Step 7

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

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

Step 8

Solve the equation for $x$ .

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Step 8.1

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

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

Step 8.2

Simplify $\pm \sqrt{\frac{9}{2}}$ .

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Step 8.2.1

Rewrite $\sqrt{\frac{9}{2}}$ as $\frac{\sqrt{9}}{\sqrt{2}}$ .

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

Step 8.2.2

Simplify the numerator.

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Step 8.2.2.1

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

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

Step 8.2.2.2

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

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

Step 8.2.3

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8.2.4

Combine and simplify the denominator.

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Step 8.2.4.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8.2.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 8.2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 8.2.4.4

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

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

Step 8.2.4.5

Add $1$ and $1$ .

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

Step 8.2.4.6

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

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Step 8.2.4.6.1

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

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

Step 8.2.4.6.2

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

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

Step 8.2.4.6.3

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

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

Step 8.2.4.6.4

Cancel the common factor of $2$ .

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Step 8.2.4.6.4.1

Cancel the common factor.

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

Step 8.2.4.6.4.2

Rewrite the expression.

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

Step 8.2.4.6.5

Evaluate the exponent.

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

Add $3$ to both sides of the equation.

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

Step 8.3.3

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

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

Step 8.3.4

Add $3$ to both sides of the equation.

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

Step 8.3.5

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

$x = \frac{3 \sqrt{2}}{2} + 3, - \frac{3 \sqrt{2}}{2} + 3$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \sqrt{2}}{2} + 3, - \frac{3 \sqrt{2}}{2} + 3$

Decimal Form:

$x = 5.12132034 \dots, 0.87867965 \dots$

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