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

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

Solve for x (complex solution)

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

$$x=-\left(\sqrt{970}+30\right)\approx -61.144823005$$

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}.

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

Square -12.

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

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

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

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

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

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

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

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

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

The opposite of -12 is 12.

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

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}.

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

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.

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

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}.

$$x^{2}+\left(-\frac{12}{-\frac{1}{5}}\right)x=-\frac{14}{-\frac{1}{5}}$$

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.

$$x^{2}+60x+30^{2}=70+30^{2}$$

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.

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

Subtract 30 from both sides of the equation.

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

Solve for x

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

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

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}.

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

Square -12.

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

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

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

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

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

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

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

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

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

The opposite of -12 is 12.

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

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}.

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

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.

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

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}.

$$x^{2}+\left(-\frac{12}{-\frac{1}{5}}\right)x=-\frac{14}{-\frac{1}{5}}$$

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.

$$x^{2}+60x+30^{2}=70+30^{2}$$

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.

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

Subtract 30 from both sides of the equation.

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