Solve x^2+24x+85=0

$${ x }^{ 2 } +24x+85=0$$

Solve for x (complex solution)

$$x=\sqrt{59}-12\approx -4.318854252$$

$$x=-\left(\sqrt{59}+12\right)\approx -19.681145748$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+24x+85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and 85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 85}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 85}}{2}$$

Multiply -4 times 85.

$$x=\frac{-24±\sqrt{576-340}}{2}$$

Add 576 to -340.

$$x=\frac{-24±\sqrt{236}}{2}$$

Take the square root of 236.

$$x=\frac{-24±2\sqrt{59}}{2}$$

Now solve the equation x=\frac{-24±2\sqrt{59}}{2} when ± is plus. Add -24 to 2\sqrt{59}.

$$x=\frac{2\sqrt{59}-24}{2}$$

Divide -24+2\sqrt{59} by 2.

$$x=\sqrt{59}-12$$

Now solve the equation x=\frac{-24±2\sqrt{59}}{2} when ± is minus. Subtract 2\sqrt{59} from -24.

$$x=\frac{-2\sqrt{59}-24}{2}$$

Divide -24-2\sqrt{59} by 2.

$$x=-\sqrt{59}-12$$

The equation is now solved.

$$x=\sqrt{59}-12 x=-\sqrt{59}-12$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+24x+85=0$$

Subtract 85 from both sides of the equation.

$$x^{2}+24x+85-85=-85$$

Subtracting 85 from itself leaves 0.

$$x^{2}+24x=-85$$

Divide 24, the coefficient of the x term, by 2 to get 12. Then add the square of 12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+24x+12^{2}=-85+12^{2}$$

Square 12.

$$x^{2}+24x+144=-85+144$$

Add -85 to 144.

$$x^{2}+24x+144=59$$

Factor x^{2}+24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+12\right)^{2}=59$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{59}$$

Simplify.

$$x+12=\sqrt{59} x+12=-\sqrt{59}$$

Subtract 12 from both sides of the equation.

$$x=\sqrt{59}-12 x=-\sqrt{59}-12$$

Solve for x

$$x=\sqrt{59}-12\approx -4.318854252$$

$$x=-\sqrt{59}-12\approx -19.681145748$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+24x+85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and 85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 85}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 85}}{2}$$

Multiply -4 times 85.

$$x=\frac{-24±\sqrt{576-340}}{2}$$

Add 576 to -340.

$$x=\frac{-24±\sqrt{236}}{2}$$

Take the square root of 236.

$$x=\frac{-24±2\sqrt{59}}{2}$$

Now solve the equation x=\frac{-24±2\sqrt{59}}{2} when ± is plus. Add -24 to 2\sqrt{59}.

$$x=\frac{2\sqrt{59}-24}{2}$$

Divide -24+2\sqrt{59} by 2.

$$x=\sqrt{59}-12$$

Now solve the equation x=\frac{-24±2\sqrt{59}}{2} when ± is minus. Subtract 2\sqrt{59} from -24.

$$x=\frac{-2\sqrt{59}-24}{2}$$

Divide -24-2\sqrt{59} by 2.

$$x=-\sqrt{59}-12$$

The equation is now solved.

$$x=\sqrt{59}-12 x=-\sqrt{59}-12$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+24x+85=0$$

Subtract 85 from both sides of the equation.

$$x^{2}+24x+85-85=-85$$

Subtracting 85 from itself leaves 0.

$$x^{2}+24x=-85$$

Divide 24, the coefficient of the x term, by 2 to get 12. Then add the square of 12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+24x+12^{2}=-85+12^{2}$$

Square 12.

$$x^{2}+24x+144=-85+144$$

Add -85 to 144.

$$x^{2}+24x+144=59$$

Factor x^{2}+24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+12\right)^{2}=59$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{59}$$

Simplify.

$$x+12=\sqrt{59} x+12=-\sqrt{59}$$

Subtract 12 from both sides of the equation.

$$x=\sqrt{59}-12 x=-\sqrt{59}-12$$