Solve Using the Quadratic Formula x^2-24x+70=-30

$x^{2} - 24 x + 70 = - 30$

Step 1

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

Add $30$ to both sides of the equation.

$x^{2} - 24 x + 70 + 30 = 0$

Add $70$ and $30$ .

$x^{2} - 24 x + 100 = 0$

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 $a = 1$ , $b = - 24$ , and $c = 100$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (1 \cdot 100)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 1 \cdot 100}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 100$ .

Multiply $- 4$ by $1$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 100}}{2 \cdot 1}$

Multiply $- 4$ by $100$ .

$x = \frac{24 \pm \sqrt{576 - 400}}{2 \cdot 1}$

Subtract $400$ from $576$ .

$x = \frac{24 \pm \sqrt{176}}{2 \cdot 1}$

Rewrite $176$ as $4^{2} \cdot 11$ .

Factor $16$ out of $176$ .

$x = \frac{24 \pm \sqrt{16 (11)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{24 \pm \sqrt{4^{2} \cdot 11}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{24 \pm 4 \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{24 \pm 4 \sqrt{11}}{2}$

Simplify $\frac{24 \pm 4 \sqrt{11}}{2}$ .

$x = 12 \pm 2 \sqrt{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 12 \pm 2 \sqrt{11}$

Decimal Form:

$x = 18.63324958 \dots, 5.36675041 \dots$