Precalculus Examples

$5 x^{2} + 20 x - 21 = 0$

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 $a = 5$ , $b = 20$ , and $c = - 21$ into the quadratic formula and solve for $x$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (5 \cdot - 21)}}{2 \cdot 5}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $20$ to the power of $2$ .

$x = \frac{- 20 \pm \sqrt{400 - 4 \cdot 5 \cdot - 21}}{2 \cdot 5}$

Step 3.1.2

Multiply $- 4 \cdot 5 \cdot - 21$ .

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Step 3.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 20 \pm \sqrt{400 - 20 \cdot - 21}}{2 \cdot 5}$

Step 3.1.2.2

Multiply $- 20$ by $- 21$ .

$x = \frac{- 20 \pm \sqrt{400 + 420}}{2 \cdot 5}$

Step 3.1.3

Add $400$ and $420$ .

$x = \frac{- 20 \pm \sqrt{820}}{2 \cdot 5}$

Step 3.1.4

Rewrite $820$ as $2^{2} \cdot 205$ .

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Step 3.1.4.1

Factor $4$ out of $820$ .

$x = \frac{- 20 \pm \sqrt{4 (205)}}{2 \cdot 5}$

Step 3.1.4.2

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

$x = \frac{- 20 \pm \sqrt{2^{2} \cdot 205}}{2 \cdot 5}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{- 20 \pm 2 \sqrt{205}}{2 \cdot 5}$

Step 3.2

Multiply $2$ by $5$ .

$x = \frac{- 20 \pm 2 \sqrt{205}}{10}$

Step 3.3

Simplify $\frac{- 20 \pm 2 \sqrt{205}}{10}$ .

$x = \frac{- 10 \pm \sqrt{205}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 10 \pm \sqrt{205}}{5}$

Decimal Form:

$x = 0.86356421 \dots, - 4.86356421 \dots$