Basic Math Examples

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

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 $a = 1$ , $b = 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

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 $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 3.1.3

Subtract $20$ from $16$ .

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

Step 3.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 3.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 3.1.6

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

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

Step 3.1.7

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

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

Step 3.1.8

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

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

Step 3.1.9

Move $2$ to the left of $i$ .

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

Step 3.2

Multiply $2$ by $1$ .

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

Step 3.3

Simplify $\frac{- 4 \pm 2 i}{2}$ .

$x = - 2 \pm i$

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 $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 4.1.3

Subtract $20$ from $16$ .

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

Step 4.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 4.1.6

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

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

Step 4.1.7

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

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

Step 4.1.8

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

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

Step 4.1.9

Move $2$ to the left of $i$ .

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

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

Simplify $\frac{- 4 \pm 2 i}{2}$ .

$x = - 2 \pm i$

Step 4.4

Change the $\pm$ to $+$ .

$x = - 2 + i$

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 $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 5.1.3

Subtract $20$ from $16$ .

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

Step 5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

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

Step 5.1.9

Move $2$ to the left of $i$ .

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

Simplify $\frac{- 4 \pm 2 i}{2}$ .

$x = - 2 \pm i$

Step 5.4

Change the $\pm$ to $-$ .

$x = - 2 - i$

Step 6

The final answer is the combination of both solutions.

$x = - 2 + i, - 2 - i$