Algebra Examples

2 x^{2} - x - 15 = 0

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

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = 2

$a = 2$ ,

b = - 1

$b = - 1$ , and

c = - 15

$c = - 15$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 3.1.2

Multiply

- 4 \cdot 2 \cdot - 15

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

Tap for more steps...

Step 3.1.2.1

Multiply

- 4

$- 4$ by

2

$2$ .

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

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

Step 3.1.2.2

Multiply

- 8

$- 8$ by

- 15

$- 15$ .

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

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

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

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

Step 3.1.3

Add

1

$1$ and

120

$120$ .

x = \frac{1 \pm \sqrt{121}}{2 \cdot 2}

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

Step 3.1.4

Rewrite

121

$121$ as

11^{2}

$11^{2}$ .

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

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

Step 3.1.5

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

x = \frac{1 \pm 11}{2 \cdot 2}

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

x = \frac{1 \pm 11}{2 \cdot 2}

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

Step 3.2

Multiply

2

$2$ by

2

$2$ .

x = \frac{1 \pm 11}{4}

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

x = \frac{1 \pm 11}{4}

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

Step 4

The final answer is the combination of both solutions.

x = 3, - \frac{5}{2}

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