Precalculus Examples

$4 x^{2} - 2 x + 11 = 0$

Step 1

Subtract $11$ from both sides of the equation.

$4 x^{2} - 2 x = - 11$

Step 2

Divide each term in $4 x^{2} - 2 x = - 11$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x^{2} - 2 x = - 11$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{- 2 x}{4} = \frac{- 11}{4}$

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.

$\frac{4 x^{2}}{4} + \frac{- 2 x}{4} = \frac{- 11}{4}$

Step 2.2.1.1.2

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

$x^{2} + \frac{- 2 x}{4} = \frac{- 11}{4}$

Step 2.2.1.2

Cancel the common factor of $- 2$ and $4$ .

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Step 2.2.1.2.1

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

$x^{2} + \frac{2 (- x)}{4} = \frac{- 11}{4}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $2$ out of $4$ .

$x^{2} + \frac{2 (- x)}{2 (2)} = \frac{- 11}{4}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{2 (- x)}{2 \cdot 2} = \frac{- 11}{4}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{- x}{2} = \frac{- 11}{4}$

Step 2.2.1.3

Move the negative in front of the fraction.

$x^{2} - \frac{x}{2} = \frac{- 11}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x^{2} - \frac{x}{2} = - \frac{11}{4}$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} - \frac{x}{2} + {(- \frac{1}{4})}^{2} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

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

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Step 5.1.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$x^{2} - \frac{x}{2} + {(- 1)}^{2} {(\frac{1}{4})}^{2} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$x^{2} - \frac{x}{2} + {(- 1)}^{2} \frac{1^{2}}{4^{2}} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.2

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

$x^{2} - \frac{x}{2} + 1 \frac{1^{2}}{4^{2}} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.3

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

$x^{2} - \frac{x}{2} + \frac{1^{2}}{4^{2}} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.4

One to any power is one.

$x^{2} - \frac{x}{2} + \frac{1}{4^{2}} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5.1.1.5

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

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + {(- \frac{1}{4})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{11}{4} + {(- \frac{1}{4})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + {(- 1)}^{2} {(\frac{1}{4})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + {(- 1)}^{2} \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.2

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

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + 1 \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.3

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

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.4

One to any power is one.

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + \frac{1}{4^{2}}$

Step 5.2.1.1.5

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

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} + \frac{1}{16}$

Step 5.2.1.2

To write $- \frac{11}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11}{4} \cdot \frac{4}{4} + \frac{1}{16}$

Step 5.2.1.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{11}{4}$ by $\frac{4}{4}$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11 \cdot 4}{4 \cdot 4} + \frac{1}{16}$

Step 5.2.1.3.2

Multiply $4$ by $4$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{11 \cdot 4}{16} + \frac{1}{16}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} - \frac{x}{2} + \frac{1}{16} = \frac{- 11 \cdot 4 + 1}{16}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $- 11$ by $4$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = \frac{- 44 + 1}{16}$

Step 5.2.1.5.2

Add $- 44$ and $1$ .

$x^{2} - \frac{x}{2} + \frac{1}{16} = \frac{- 43}{16}$

Step 5.2.1.6

Move the negative in front of the fraction.

$x^{2} - \frac{x}{2} + \frac{1}{16} = - \frac{43}{16}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{1}{4})}^{2}$ .

${(x - \frac{1}{4})}^{2} = - \frac{43}{16}$

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.

$x - \frac{1}{4} = \pm \sqrt{- \frac{43}{16}}$

Step 7.2

Simplify $\pm \sqrt{- \frac{43}{16}}$ .

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Step 7.2.1

Rewrite $- \frac{43}{16}$ as ${(\frac{i}{4})}^{2} \cdot 43$ .

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Step 7.2.1.1

Rewrite $- 1$ as $i^{2}$ .

$x - \frac{1}{4} = \pm \sqrt{i^{2} \frac{43}{16}}$

Step 7.2.1.2

Factor the perfect power $1^{2}$ out of $43$ .

$x - \frac{1}{4} = \pm \sqrt{i^{2} \frac{1^{2} \cdot 43}{16}}$

Step 7.2.1.3

Factor the perfect power $4^{2}$ out of $16$ .

$x - \frac{1}{4} = \pm \sqrt{i^{2} \frac{1^{2} \cdot 43}{4^{2} \cdot 1}}$

Step 7.2.1.4

Rearrange the fraction $\frac{1^{2} \cdot 43}{4^{2} \cdot 1}$ .

$x - \frac{1}{4} = \pm \sqrt{i^{2} ({(\frac{1}{4})}^{2} \cdot 43)}$

Step 7.2.1.5

Rewrite $i^{2} {(\frac{1}{4})}^{2}$ as ${(\frac{i}{4})}^{2}$ .

$x - \frac{1}{4} = \pm \sqrt{{(\frac{i}{4})}^{2} \cdot 43}$

Step 7.2.2

Pull terms out from under the radical.

$x - \frac{1}{4} = \pm \frac{i}{4} \sqrt{43}$

Step 7.2.3

Combine $\frac{i}{4}$ and $\sqrt{43}$ .

$x - \frac{1}{4} = \pm \frac{i \sqrt{43}}{4}$

Step 7.3

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.1

Add $\frac{1}{4}$ to both sides of the equation.

$x = \pm \frac{i \sqrt{43}}{4} + \frac{1}{4}$

Step 7.3.2

Reorder $\pm \frac{i \sqrt{43}}{4}$ and $\frac{1}{4}$ .

$x = \frac{1}{4} \pm \frac{i \sqrt{43}}{4}$