Pre-Algebra Examples

$- 4 x^{2} - 4 x - 5 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = - 4$ , $b = - 4$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 4 \cdot - 5)}}{2 \cdot - 4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4 \cdot - 5}}{2 \cdot - 4}$

Step 3.1.2

Multiply $- 4 \cdot - 4 \cdot - 5$ .

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Step 3.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16 \cdot - 5}}{2 \cdot - 4}$

Step 3.1.2.2

Multiply $16$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot - 4}$

Step 3.1.3

Subtract $80$ from $16$ .

$x = \frac{4 \pm \sqrt{- 64}}{2 \cdot - 4}$

Step 3.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 64}}{2 \cdot - 4}$

Step 3.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot - 4}$

Step 3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot - 4}$

Step 3.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot - 4}$

Step 3.1.8

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot - 4}$

Step 3.1.9

Move $8$ to the left of $i$ .

$x = \frac{4 \pm 8 i}{2 \cdot - 4}$

Step 3.2

Multiply $2$ by $- 4$ .

$x = \frac{4 \pm 8 i}{- 8}$

Step 3.3

Simplify $\frac{4 \pm 8 i}{- 8}$ .

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

Step 3.4

Move the negative in front of the fraction.

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

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4 \cdot - 5}}{2 \cdot - 4}$

Step 4.1.2

Multiply $- 4 \cdot - 4 \cdot - 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16 \cdot - 5}}{2 \cdot - 4}$

Step 4.1.2.2

Multiply $16$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot - 4}$

Step 4.1.3

Subtract $80$ from $16$ .

$x = \frac{4 \pm \sqrt{- 64}}{2 \cdot - 4}$

Step 4.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 64}}{2 \cdot - 4}$

Step 4.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot - 4}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot - 4}$

Step 4.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot - 4}$

Step 4.1.8

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot - 4}$

Step 4.1.9

Move $8$ to the left of $i$ .

$x = \frac{4 \pm 8 i}{2 \cdot - 4}$

Step 4.2

Multiply $2$ by $- 4$ .

$x = \frac{4 \pm 8 i}{- 8}$

Step 4.3

Simplify $\frac{4 \pm 8 i}{- 8}$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 4.5

Change the $\pm$ to $+$ .

$x = - \frac{1 + 2 i}{2}$

Step 4.6

Split the fraction $\frac{1 + 2 i}{2}$ into two fractions.

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

Step 4.7

Cancel the common factor of $2$ .

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Step 4.7.1

Cancel the common factor.

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

Step 4.7.2

Divide $i$ by $1$ .

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

Step 4.8

Apply the distributive property.

$x = - \frac{1}{2} - i$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4 \cdot - 5}}{2 \cdot - 4}$

Step 5.1.2

Multiply $- 4 \cdot - 4 \cdot - 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16 \cdot - 5}}{2 \cdot - 4}$

Step 5.1.2.2

Multiply $16$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot - 4}$

Step 5.1.3

Subtract $80$ from $16$ .

$x = \frac{4 \pm \sqrt{- 64}}{2 \cdot - 4}$

Step 5.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 64}}{2 \cdot - 4}$

Step 5.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot - 4}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot - 4}$

Step 5.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot - 4}$

Step 5.1.8

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

$x = \frac{4 \pm i \cdot 8}{2 \cdot - 4}$

Step 5.1.9

Move $8$ to the left of $i$ .

$x = \frac{4 \pm 8 i}{2 \cdot - 4}$

Step 5.2

Multiply $2$ by $- 4$ .

$x = \frac{4 \pm 8 i}{- 8}$

Step 5.3

Simplify $\frac{4 \pm 8 i}{- 8}$ .

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

Step 5.4

Move the negative in front of the fraction.

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

Step 5.5

Change the $\pm$ to $-$ .

$x = - \frac{1 - 2 i}{2}$

Step 5.6

Split the fraction $\frac{1 - 2 i}{2}$ into two fractions.

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

Step 5.7

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

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Step 5.7.1

Factor $2$ out of $- 2 i$ .

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

Step 5.7.2

Cancel the common factors.

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Step 5.7.2.1

Factor $2$ out of $2$ .

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

Step 5.7.2.2

Cancel the common factor.

$x = - (\frac{1}{2} + \frac{2 (- i)}{2 \cdot 1})$

Step 5.7.2.3

Rewrite the expression.

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

Step 5.7.2.4

Divide $- i$ by $1$ .

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

Step 5.8

Apply the distributive property.

$x = - \frac{1}{2} + i$

Step 5.9

Multiply $- 1$ by $- 1$ .

$x = - \frac{1}{2} + 1 i$

Step 5.10

Multiply $i$ by $1$ .

$x = - \frac{1}{2} + i$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{1}{2} - i, - \frac{1}{2} + i$