Precalculus Examples

$4 x^{2} - 16 x + 9 = 0$

Step 1

Subtract $9$ from both sides of the equation.

$4 x^{2} - 16 x = - 9$

Step 2

Divide each term in $4 x^{2} - 16 x = - 9$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x^{2} - 16 x = - 9$ by $4$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Cancel the common factor of $- 16$ and $4$ .

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Step 2.2.1.2.1

Factor $4$ out of $- 16 x$ .

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

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $4$ out of $4$ .

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

Step 2.2.1.2.2.2

Cancel the common factor.

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

Step 2.2.1.2.2.3

Rewrite the expression.

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

Step 2.2.1.2.2.4

Divide $- 4 x$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Combine the numerators over the common denominator.

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

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $4$ by $4$ .

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

Step 5.2.1.5.2

Add $- 9$ and $16$ .

$x^{2} - 4 x + 4 = \frac{7}{4}$

Step 6

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

${(x - 2)}^{2} = \frac{7}{4}$

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.

$x - 2 = \pm \sqrt{\frac{7}{4}}$

Step 7.2

Simplify $\pm \sqrt{\frac{7}{4}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{7}{4}}$ as $\frac{\sqrt{7}}{\sqrt{4}}$ .

$x - 2 = \pm \frac{\sqrt{7}}{\sqrt{4}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$x - 2 = \pm \frac{\sqrt{7}}{\sqrt{2^{2}}}$

Step 7.2.2.2

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

$x - 2 = \pm \frac{\sqrt{7}}{2}$

Step 7.3

Add $2$ to both sides of the equation.

$x = \pm \frac{\sqrt{7}}{2} + 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{7}}{2} + 2$

Decimal Form:

$x = 3.32287565 \dots, 0.67712434 \dots$