Algebra Examples

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

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

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = 5

$a = 5$ ,

b = 5

$b = 5$ , and

c = 4

$c = 4$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

5

$5$ to the power of

2

$2$ .

x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 5 \cdot 4}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 5 \cdot 4}}{2 \cdot 5}$

Step 3.1.2

Multiply

- 4 \cdot 5 \cdot 4

$- 4 \cdot 5 \cdot 4$ .

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

5

$5$ .

x = \frac{- 5 \pm \sqrt{25 - 20 \cdot 4}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{25 - 20 \cdot 4}}{2 \cdot 5}$

Step 3.1.2.2

Multiply

- 20

$- 20$ by

4

$4$ .

x = \frac{- 5 \pm \sqrt{25 - 80}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{25 - 80}}{2 \cdot 5}$

x = \frac{- 5 \pm \sqrt{25 - 80}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{25 - 80}}{2 \cdot 5}$

Step 3.1.3

Subtract

80

$80$ from

25

$25$ .

x = \frac{- 5 \pm \sqrt{- 55}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{- 55}}{2 \cdot 5}$

Step 3.1.4

Rewrite

- 55

$- 55$ as

- 1 (55)

$- 1 (55)$ .

x = \frac{- 5 \pm \sqrt{- 1 \cdot 55}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 55}}{2 \cdot 5}$

Step 3.1.5

Rewrite

\sqrt{- 1 (55)}

$\sqrt{- 1 (55)}$ as

\sqrt{- 1} \cdot \sqrt{55}

$\sqrt{- 1} \cdot \sqrt{55}$ .

x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot 5}

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot 5}$

Step 3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

x = \frac{- 5 \pm i \sqrt{55}}{2 \cdot 5}

$x = \frac{- 5 \pm i \sqrt{55}}{2 \cdot 5}$

x = \frac{- 5 \pm i \sqrt{55}}{2 \cdot 5}

$x = \frac{- 5 \pm i \sqrt{55}}{2 \cdot 5}$

Step 3.2

Multiply

2

$2$ by

5

$5$ .

x = \frac{- 5 \pm i \sqrt{55}}{10}

$x = \frac{- 5 \pm i \sqrt{55}}{10}$

x = \frac{- 5 \pm i \sqrt{55}}{10}

$x = \frac{- 5 \pm i \sqrt{55}}{10}$