Solve Using the Quadratic Formula 2a^2-a-13=2

$2 a^{2} - a - 13 = 2$

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $2$ from both sides of the equation.

$2 a^{2} - a - 13 - 2 = 0$

Subtract $2$ from $- 13$ .

$2 a^{2} - a - 15 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 2$ , $b = - 1$ , and $c = - 15$ into the quadratic formula and solve for $a$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot - 15)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$a = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 15}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 15$ .

Multiply $- 4$ by $2$ .

$a = \frac{1 \pm \sqrt{1 - 8 \cdot - 15}}{2 \cdot 2}$

Multiply $- 8$ by $- 15$ .

$a = \frac{1 \pm \sqrt{1 + 120}}{2 \cdot 2}$

Add $1$ and $120$ .

$a = \frac{1 \pm \sqrt{121}}{2 \cdot 2}$

Rewrite $121$ as $11^{2}$ .

$a = \frac{1 \pm \sqrt{11^{2}}}{2 \cdot 2}$

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

$a = \frac{1 \pm 11}{2 \cdot 2}$

Multiply $2$ by $2$ .

$a = \frac{1 \pm 11}{4}$

Step 5

The final answer is the combination of both solutions.

$a = 3, - \frac{5}{2}$