Solve 18=-1/5x^2+12x+32

$$18 = - \frac { 1 } { 5 } x ^ { 2 } + 12 x + 32$$

Solve for x

$$x=\sqrt{970}+30\approx 61.144823005$$

$$x=30-\sqrt{970}\approx -1.144823005$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{5}x^{2}+12x+32=18$$

Subtract 18 from both sides.

$$-\frac{1}{5}x^{2}+12x+32-18=0$$

Subtract 18 from 32 to get 14.

$$-\frac{1}{5}x^{2}+12x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{5} for a, 12 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

Multiply -4 times -\frac{1}{5}.

$$x=\frac{-12±\sqrt{144+\frac{4}{5}\times 14}}{2\left(-\frac{1}{5}\right)}$$

Multiply \frac{4}{5} times 14.

$$x=\frac{-12±\sqrt{144+\frac{56}{5}}}{2\left(-\frac{1}{5}\right)}$$

Add 144 to \frac{56}{5}.

$$x=\frac{-12±\sqrt{\frac{776}{5}}}{2\left(-\frac{1}{5}\right)}$$

Take the square root of \frac{776}{5}.

$$x=\frac{-12±\frac{2\sqrt{970}}{5}}{2\left(-\frac{1}{5}\right)}$$

Multiply 2 times -\frac{1}{5}.

$$x=\frac{-12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}}$$

Now solve the equation x=\frac{-12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} when ± is plus. Add -12 to \frac{2\sqrt{970}}{5}.

$$x=\frac{\frac{2\sqrt{970}}{5}-12}{-\frac{2}{5}}$$

Divide -12+\frac{2\sqrt{970}}{5} by -\frac{2}{5} by multiplying -12+\frac{2\sqrt{970}}{5} by the reciprocal of -\frac{2}{5}.

$$x=30-\sqrt{970}$$

Now solve the equation x=\frac{-12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} when ± is minus. Subtract \frac{2\sqrt{970}}{5} from -12.

$$x=\frac{-\frac{2\sqrt{970}}{5}-12}{-\frac{2}{5}}$$

Divide -12-\frac{2\sqrt{970}}{5} by -\frac{2}{5} by multiplying -12-\frac{2\sqrt{970}}{5} by the reciprocal of -\frac{2}{5}.

$$x=\sqrt{970}+30$$

The equation is now solved.

$$x=30-\sqrt{970} x=\sqrt{970}+30$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{5}x^{2}+12x+32=18$$

Subtract 32 from both sides.

$$-\frac{1}{5}x^{2}+12x=18-32$$

Subtract 32 from 18 to get -14.

$$-\frac{1}{5}x^{2}+12x=-14$$

Multiply both sides by -5.

$$\frac{-\frac{1}{5}x^{2}+12x}{-\frac{1}{5}}=-\frac{14}{-\frac{1}{5}}$$

Dividing by -\frac{1}{5} undoes the multiplication by -\frac{1}{5}.

$$x^{2}+\frac{12}{-\frac{1}{5}}x=-\frac{14}{-\frac{1}{5}}$$

Divide 12 by -\frac{1}{5} by multiplying 12 by the reciprocal of -\frac{1}{5}.

$$x^{2}-60x=-\frac{14}{-\frac{1}{5}}$$

Divide -14 by -\frac{1}{5} by multiplying -14 by the reciprocal of -\frac{1}{5}.

$$x^{2}-60x=70$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=70+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=70+900$$

Add 70 to 900.

$$x^{2}-60x+900=970$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=970$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{970}$$

Simplify.

$$x-30=\sqrt{970} x-30=-\sqrt{970}$$

Add 30 to both sides of the equation.

$$x=\sqrt{970}+30 x=30-\sqrt{970}$$