Finite Math Examples Equations and Inequalities

Solve by Completing the Square

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

Step 1

Add $3$ to both sides of the equation.

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

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 $b$ .

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

Step 3

Add the term to each side of the equation.

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

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

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

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Step 4.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

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

Step 4.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

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

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

Step 4.1.1.5

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

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

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

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Step 4.2.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

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

Step 4.2.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

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

Step 4.2.1.1.2

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

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

Step 4.2.1.1.3

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

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

Step 4.2.1.1.4

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

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

Step 4.2.1.1.5

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

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

Step 4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 4.2.1.3

Combine $3$ and $\frac{4}{4}$ .

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

Step 4.2.1.4

Combine the numerators over the common denominator.

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

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $3$ by $4$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{12 + 9}{4}$

Step 4.2.1.5.2

Add $12$ and $9$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{21}{4}$

Step 5

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

${(x - \frac{3}{2})}^{2} = \frac{21}{4}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x - \frac{3}{2} = \pm \sqrt{\frac{21}{4}}$

Step 6.2

Simplify $\pm \sqrt{\frac{21}{4}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{21}{4}}$ as $\frac{\sqrt{21}}{\sqrt{4}}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{21}}{\sqrt{4}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

$x - \frac{3}{2} = \pm \frac{\sqrt{21}}{\sqrt{2^{2}}}$

Step 6.2.2.2

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

$x - \frac{3}{2} = \pm \frac{\sqrt{21}}{2}$

Step 6.3

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

$x = \pm \frac{\sqrt{21}}{2} + \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{21}}{2} + \frac{3}{2}$

Decimal Form:

$x = 3.79128784 \dots, - 0.79128784 \dots$

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