Algebra Examples

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

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

$a = 4$ ,

$b = - 1$ , and

$c = - 5$ into the quadratic formula and solve for

$x$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

$- 1$ to the power of

$2$ .

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

Step 3.1.2

Multiply

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

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Step 3.1.2.1

Multiply

$- 4$ by

$4$ .

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

Step 3.1.2.2

Multiply

$- 16$ by

$- 5$ .

$x = \frac{1 \pm \sqrt{1 + 80}}{2 \cdot 4}$

Step 3.1.3

Add

$1$ and

$80$ .

$x = \frac{1 \pm \sqrt{81}}{2 \cdot 4}$

Step 3.1.4

Rewrite

$81$ as

$9^{2}$ .

$x = \frac{1 \pm \sqrt{9^{2}}}{2 \cdot 4}$

Step 3.1.5

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

$x = \frac{1 \pm 9}{2 \cdot 4}$

Step 3.2

Multiply

$2$ by

$4$ .

$x = \frac{1 \pm 9}{8}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{5}{4}, - 1$