Solve 75=4x-1/2x^2

$$75 = 4 x - \frac { 1 } { 2 } x ^ { 2 }$$

Solve for x (complex solution)

$$x=4+\sqrt{134}i\approx 4+11.575836903i$$

$$x=-\sqrt{134}i+4\approx 4-11.575836903i$$

Steps Using the Quadratic Formula

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

$$4x-\frac{1}{2}x^{2}=75$$

Subtract 75 from both sides.

$$4x-\frac{1}{2}x^{2}-75=0$$

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.

$$-\frac{1}{2}x^{2}+4x-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, 4 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-\frac{1}{2}\right)\left(-75\right)}}{2\left(-\frac{1}{2}\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-\frac{1}{2}\right)\left(-75\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply -4 times -\frac{1}{2}.

$$x=\frac{-4±\sqrt{16+2\left(-75\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -75.

$$x=\frac{-4±\sqrt{16-150}}{2\left(-\frac{1}{2}\right)}$$

Add 16 to -150.

$$x=\frac{-4±\sqrt{-134}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of -134.

$$x=\frac{-4±\sqrt{134}i}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-4±\sqrt{134}i}{-1}$$

Now solve the equation x=\frac{-4±\sqrt{134}i}{-1} when ± is plus. Add -4 to i\sqrt{134}.

$$x=\frac{-4+\sqrt{134}i}{-1}$$

Divide -4+i\sqrt{134} by -1.

$$x=-\sqrt{134}i+4$$

Now solve the equation x=\frac{-4±\sqrt{134}i}{-1} when ± is minus. Subtract i\sqrt{134} from -4.

$$x=\frac{-\sqrt{134}i-4}{-1}$$

Divide -4-i\sqrt{134} by -1.

$$x=4+\sqrt{134}i$$

The equation is now solved.

$$x=-\sqrt{134}i+4 x=4+\sqrt{134}i$$

Steps for Completing the Square

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

$$4x-\frac{1}{2}x^{2}=75$$

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.

$$-\frac{1}{2}x^{2}+4x=75$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}x^{2}+4x}{-\frac{1}{2}}=\frac{75}{-\frac{1}{2}}$$

Dividing by -\frac{1}{2} undoes the multiplication by -\frac{1}{2}.

$$x^{2}+\frac{4}{-\frac{1}{2}}x=\frac{75}{-\frac{1}{2}}$$

Divide 4 by -\frac{1}{2} by multiplying 4 by the reciprocal of -\frac{1}{2}.

$$x^{2}-8x=\frac{75}{-\frac{1}{2}}$$

Divide 75 by -\frac{1}{2} by multiplying 75 by the reciprocal of -\frac{1}{2}.

$$x^{2}-8x=-150$$

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

$$x^{2}-8x+\left(-4\right)^{2}=-150+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-150+16$$

Add -150 to 16.

$$x^{2}-8x+16=-134$$

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

$$\left(x-4\right)^{2}=-134$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-134}$$

Simplify.

$$x-4=\sqrt{134}i x-4=-\sqrt{134}i$$

Add 4 to both sides of the equation.

$$x=4+\sqrt{134}i x=-\sqrt{134}i+4$$