Solve 112=6x-1/2*75x^2

$$112 = 6 x - \frac { 1 } { 2 } \times 75 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}\approx 0.08+1.726344886i$$

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}\approx 0.08-1.726344886i$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 75 to get \frac{75}{2}.

$$112=6x-\frac{75}{2}x^{2}$$

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

$$6x-\frac{75}{2}x^{2}=112$$

Subtract 112 from both sides.

$$6x-\frac{75}{2}x^{2}-112=0$$

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.

$$-\frac{75}{2}x^{2}+6x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{75}{2} for a, 6 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-\frac{75}{2}\right)\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-\frac{75}{2}\right)\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

Multiply -4 times -\frac{75}{2}.

$$x=\frac{-6±\sqrt{36+150\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

Multiply 150 times -112.

$$x=\frac{-6±\sqrt{36-16800}}{2\left(-\frac{75}{2}\right)}$$

Add 36 to -16800.

$$x=\frac{-6±\sqrt{-16764}}{2\left(-\frac{75}{2}\right)}$$

Take the square root of -16764.

$$x=\frac{-6±2\sqrt{4191}i}{2\left(-\frac{75}{2}\right)}$$

Multiply 2 times -\frac{75}{2}.

$$x=\frac{-6±2\sqrt{4191}i}{-75}$$

Now solve the equation x=\frac{-6±2\sqrt{4191}i}{-75} when ± is plus. Add -6 to 2i\sqrt{4191}.

$$x=\frac{-6+2\sqrt{4191}i}{-75}$$

Divide -6+2i\sqrt{4191} by -75.

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Now solve the equation x=\frac{-6±2\sqrt{4191}i}{-75} when ± is minus. Subtract 2i\sqrt{4191} from -6.

$$x=\frac{-2\sqrt{4191}i-6}{-75}$$

Divide -6-2i\sqrt{4191} by -75.

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

The equation is now solved.

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25} x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 75 to get \frac{75}{2}.

$$112=6x-\frac{75}{2}x^{2}$$

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

$$6x-\frac{75}{2}x^{2}=112$$

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.

$$-\frac{75}{2}x^{2}+6x=112$$

Divide both sides of the equation by -\frac{75}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{75}{2}x^{2}+6x}{-\frac{75}{2}}=\frac{112}{-\frac{75}{2}}$$

Dividing by -\frac{75}{2} undoes the multiplication by -\frac{75}{2}.

$$x^{2}+\frac{6}{-\frac{75}{2}}x=\frac{112}{-\frac{75}{2}}$$

Divide 6 by -\frac{75}{2} by multiplying 6 by the reciprocal of -\frac{75}{2}.

$$x^{2}-\frac{4}{25}x=\frac{112}{-\frac{75}{2}}$$

Divide 112 by -\frac{75}{2} by multiplying 112 by the reciprocal of -\frac{75}{2}.

$$x^{2}-\frac{4}{25}x=-\frac{224}{75}$$

Divide -\frac{4}{25}, the coefficient of the x term, by 2 to get -\frac{2}{25}. Then add the square of -\frac{2}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{25}x+\left(-\frac{2}{25}\right)^{2}=-\frac{224}{75}+\left(-\frac{2}{25}\right)^{2}$$

Square -\frac{2}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{25}x+\frac{4}{625}=-\frac{224}{75}+\frac{4}{625}$$

Add -\frac{224}{75} to \frac{4}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{25}x+\frac{4}{625}=-\frac{5588}{1875}$$

Factor x^{2}-\frac{4}{25}x+\frac{4}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{25}\right)^{2}=-\frac{5588}{1875}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{25}\right)^{2}}=\sqrt{-\frac{5588}{1875}}$$

Simplify.

$$x-\frac{2}{25}=\frac{2\sqrt{4191}i}{75} x-\frac{2}{25}=-\frac{2\sqrt{4191}i}{75}$$

Add \frac{2}{25} to both sides of the equation.

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25} x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$