Precalculus Examples

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

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 $a = 2$ , $b = 1$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot - 5}}{2 \cdot 2}$

Step 3.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 40}}{2 \cdot 2}$

Step 3.1.3

Add $1$ and $40$ .

$x = \frac{- 1 \pm \sqrt{41}}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm \sqrt{41}}{4}$

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

One to any power is one.

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

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot - 5}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 40}}{2 \cdot 2}$

Step 4.1.3

Add $1$ and $40$ .

$x = \frac{- 1 \pm \sqrt{41}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm \sqrt{41}}{4}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{41}}{4}$

Step 4.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{41}}{4}$

Step 4.5

Factor $- 1$ out of $\sqrt{41}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{41})}{4}$

Step 4.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{41})$ .

$x = \frac{- 1 (1 - \sqrt{41})}{4}$

Step 4.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{41}}{4}$

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

One to any power is one.

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

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot - 5}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 40}}{2 \cdot 2}$

Step 5.1.3

Add $1$ and $40$ .

$x = \frac{- 1 \pm \sqrt{41}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm \sqrt{41}}{4}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{41}}{4}$

Step 5.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{41}}{4}$

Step 5.5

Factor $- 1$ out of $- \sqrt{41}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{41})}{4}$

Step 5.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{41})$ .

$x = \frac{- 1 (1 + \sqrt{41})}{4}$

Step 5.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{41}}{4}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{41}}{4}, - \frac{1 + \sqrt{41}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{41}}{4}, - \frac{1 + \sqrt{41}}{4}$

Decimal Form:

$x = 1.35078105 \dots, - 1.85078105 \dots$