Solve 11.2=6x-1/2*7.5x^2

$$11.2 = 6 x - \frac { 1 } { 2 } \times 7.5 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{i\times 4\sqrt{33}}{15}+0.8\approx 0.8+1.531883372i$$

$$x=-\frac{i\times 4\sqrt{33}}{15}+0.8\approx 0.8-1.531883372i$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 7.5 to get \frac{15}{4}.

$$11.2=6x-\frac{15}{4}x^{2}$$

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

$$6x-\frac{15}{4}x^{2}=11.2$$

Subtract 11.2 from both sides.

$$6x-\frac{15}{4}x^{2}-11.2=0$$

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.

$$-\frac{15}{4}x^{2}+6x-11.2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{15}{4} for a, 6 for b, and -11.2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-\frac{15}{4}\right)\left(-11.2\right)}}{2\left(-\frac{15}{4}\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-\frac{15}{4}\right)\left(-11.2\right)}}{2\left(-\frac{15}{4}\right)}$$

Multiply -4 times -\frac{15}{4}.

$$x=\frac{-6±\sqrt{36+15\left(-11.2\right)}}{2\left(-\frac{15}{4}\right)}$$

Multiply 15 times -11.2.

$$x=\frac{-6±\sqrt{36-168}}{2\left(-\frac{15}{4}\right)}$$

Add 36 to -168.

$$x=\frac{-6±\sqrt{-132}}{2\left(-\frac{15}{4}\right)}$$

Take the square root of -132.

$$x=\frac{-6±2\sqrt{33}i}{2\left(-\frac{15}{4}\right)}$$

Multiply 2 times -\frac{15}{4}.

$$x=\frac{-6±2\sqrt{33}i}{-\frac{15}{2}}$$

Now solve the equation x=\frac{-6±2\sqrt{33}i}{-\frac{15}{2}} when ± is plus. Add -6 to 2i\sqrt{33}.

$$x=\frac{-6+2\sqrt{33}i}{-\frac{15}{2}}$$

Divide -6+2i\sqrt{33} by -\frac{15}{2} by multiplying -6+2i\sqrt{33} by the reciprocal of -\frac{15}{2}.

$$x=-\frac{4\sqrt{33}i}{15}+\frac{4}{5}$$

Now solve the equation x=\frac{-6±2\sqrt{33}i}{-\frac{15}{2}} when ± is minus. Subtract 2i\sqrt{33} from -6.

$$x=\frac{-2\sqrt{33}i-6}{-\frac{15}{2}}$$

Divide -6-2i\sqrt{33} by -\frac{15}{2} by multiplying -6-2i\sqrt{33} by the reciprocal of -\frac{15}{2}.

$$x=\frac{4\sqrt{33}i}{15}+\frac{4}{5}$$

The equation is now solved.

$$x=-\frac{4\sqrt{33}i}{15}+\frac{4}{5} x=\frac{4\sqrt{33}i}{15}+\frac{4}{5}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 7.5 to get \frac{15}{4}.

$$11.2=6x-\frac{15}{4}x^{2}$$

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

$$6x-\frac{15}{4}x^{2}=11.2$$

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.

$$-\frac{15}{4}x^{2}+6x=11.2$$

Divide both sides of the equation by -\frac{15}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{15}{4}x^{2}+6x}{-\frac{15}{4}}=\frac{11.2}{-\frac{15}{4}}$$

Dividing by -\frac{15}{4} undoes the multiplication by -\frac{15}{4}.

$$x^{2}+\frac{6}{-\frac{15}{4}}x=\frac{11.2}{-\frac{15}{4}}$$

Divide 6 by -\frac{15}{4} by multiplying 6 by the reciprocal of -\frac{15}{4}.

$$x^{2}-\frac{8}{5}x=\frac{11.2}{-\frac{15}{4}}$$

Divide 11.2 by -\frac{15}{4} by multiplying 11.2 by the reciprocal of -\frac{15}{4}.

$$x^{2}-\frac{8}{5}x=-\frac{224}{75}$$

Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{224}{75}+\left(-\frac{4}{5}\right)^{2}$$

Square -\frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{224}{75}+\frac{16}{25}$$

Add -\frac{224}{75} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{176}{75}$$

Factor x^{2}-\frac{8}{5}x+\frac{16}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{5}\right)^{2}=-\frac{176}{75}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{-\frac{176}{75}}$$

Simplify.

$$x-\frac{4}{5}=\frac{4\sqrt{33}i}{15} x-\frac{4}{5}=-\frac{4\sqrt{33}i}{15}$$

Add \frac{4}{5} to both sides of the equation.

$$x=\frac{4\sqrt{33}i}{15}+\frac{4}{5} x=-\frac{4\sqrt{33}i}{15}+\frac{4}{5}$$