Solve Using the Quadratic Formula 15x^2+23x-41=0

$15 x^{2} + 23 x - 41 = 0$

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 $a = 15$ , $b = 23$ , and $c = - 41$ into the quadratic formula and solve for $x$ .

$\frac{- 23 \pm \sqrt{23^{2} - 4 \cdot (15 \cdot - 41)}}{2 \cdot 15}$

Step 3

Simplify.

Simplify the numerator.

Raise $23$ to the power of $2$ .

$x = \frac{- 23 \pm \sqrt{529 - 4 \cdot 15 \cdot - 41}}{2 \cdot 15}$

Multiply $- 4 \cdot 15 \cdot - 41$ .

Multiply $- 4$ by $15$ .

$x = \frac{- 23 \pm \sqrt{529 - 60 \cdot - 41}}{2 \cdot 15}$

Multiply $- 60$ by $- 41$ .

$x = \frac{- 23 \pm \sqrt{529 + 2460}}{2 \cdot 15}$

Add $529$ and $2460$ .

$x = \frac{- 23 \pm \sqrt{2989}}{2 \cdot 15}$

Rewrite $2989$ as $7^{2} \cdot 61$ .

Factor $49$ out of $2989$ .

$x = \frac{- 23 \pm \sqrt{49 (61)}}{2 \cdot 15}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 23 \pm \sqrt{7^{2} \cdot 61}}{2 \cdot 15}$

Pull terms out from under the radical.

$x = \frac{- 23 \pm 7 \sqrt{61}}{2 \cdot 15}$

Multiply $2$ by $15$ .

$x = \frac{- 23 \pm 7 \sqrt{61}}{30}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 23 \pm 7 \sqrt{61}}{30}$

Decimal Form:

$x = 1.05572492 \dots, - 2.58905825 \dots$