Precalculus Examples

$4 x^{2} - 16 x + 14 = 0$

Step 1

Subtract $14$ from both sides of the equation.

$4 x^{2} - 16 x = - 14$

Step 2

Divide each term in $4 x^{2} - 16 x = - 14$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x^{2} - 16 x = - 14$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{- 16 x}{4} = \frac{- 14}{4}$

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.

$\frac{4 x^{2}}{4} + \frac{- 16 x}{4} = \frac{- 14}{4}$

Step 2.2.1.1.2

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

$x^{2} + \frac{- 16 x}{4} = \frac{- 14}{4}$

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 $- 16 x$ .

$x^{2} + \frac{4 (- 4 x)}{4} = \frac{- 14}{4}$

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 $4$ .

$x^{2} + \frac{4 (- 4 x)}{4 (1)} = \frac{- 14}{4}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{4 (- 4 x)}{4 \cdot 1} = \frac{- 14}{4}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{- 4 x}{1} = \frac{- 14}{4}$

Step 2.2.1.2.2.4

Divide $- 4 x$ by $1$ .

$x^{2} - 4 x = \frac{- 14}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $- 14$ and $4$ .

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Step 2.3.1.1

Factor $2$ out of $- 14$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Move the negative in front of the fraction.

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

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.

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

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 $2$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{7}{2} + {(- 2)}^{2}$ .

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Step 5.2.1.1

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

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

Step 5.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.2.1.3

Combine $4$ and $\frac{2}{2}$ .

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

Step 5.2.1.4

Combine the numerators over the common denominator.

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

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $4$ by $2$ .

$x^{2} - 4 x + 4 = \frac{- 7 + 8}{2}$

Step 5.2.1.5.2

Add $- 7$ and $8$ .

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

Step 6

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

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

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.

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

Step 7.2

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 7.2.2

Any root of $1$ is $1$ .

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

Step 7.2.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.2.4

Combine and simplify the denominator.

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Step 7.2.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.2.4.2

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

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

Step 7.2.4.3

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

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

Step 7.2.4.4

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

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

Step 7.2.4.5

Add $1$ and $1$ .

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

Step 7.2.4.6

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

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Step 7.2.4.6.1

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

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

Step 7.2.4.6.2

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

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

Step 7.2.4.6.3

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

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

Step 7.2.4.6.4

Cancel the common factor of $2$ .

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Step 7.2.4.6.4.1

Cancel the common factor.

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

Step 7.2.4.6.4.2

Rewrite the expression.

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

Step 7.2.4.6.5

Evaluate the exponent.

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

Step 7.3

Add $2$ to both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.70710678 \dots, 1.29289321 \dots$