Solve by Factoring 2x^2+5x-1=0

$2 x^{2} + 5 x - 1 = 0$

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

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

Step 3

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot - 1}}{2 \cdot 2}$

Multiply $- 8$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 8}}{2 \cdot 2}$

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{33}}{4}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot - 1}}{2 \cdot 2}$

Multiply $- 8$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 8}}{2 \cdot 2}$

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{33}}{4}$

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{33}}{4}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{33}}{4}$

Factor $- 1$ out of $\sqrt{33}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{33})}{4}$

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{33})$ .

$x = \frac{- 1 (5 - \sqrt{33})}{4}$

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{33}}{4}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot - 1}}{2 \cdot 2}$

Multiply $- 8$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 8}}{2 \cdot 2}$

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{33}}{4}$

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{33}}{4}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{33}}{4}$

Factor $- 1$ out of $- \sqrt{33}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{33})}{4}$

Factor $- 1$ out of $- 1 (5) - (\sqrt{33})$ .

$x = \frac{- 1 (5 + \sqrt{33})}{4}$

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{33}}{4}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{33}}{4}, - \frac{5 + \sqrt{33}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 - \sqrt{33}}{4}, - \frac{5 + \sqrt{33}}{4}$

Decimal Form:

$x = 0.18614066 \dots, - 2.68614066 \dots$

Solve by Factoring 2x^2+5x-1=0

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