Solve 15x-1/2x^2-80=0

$$15 x - \frac { 1 } { 2 } x ^ { 2 } - 80 = 0$$

Solve for x

$$x=\sqrt{65}+15\approx 23.062257748$$

$$x=15-\sqrt{65}\approx 6.937742252$$

Steps Using the Quadratic Formula

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.

$$-\frac{1}{2}x^{2}+15x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, 15 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-\frac{1}{2}\right)\left(-80\right)}}{2\left(-\frac{1}{2}\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-\frac{1}{2}\right)\left(-80\right)}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-15±\sqrt{225+2\left(-80\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -80.

$$x=\frac{-15±\sqrt{225-160}}{2\left(-\frac{1}{2}\right)}$$

Add 225 to -160.

$$x=\frac{-15±\sqrt{65}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-15±\sqrt{65}}{-1}$$

Now solve the equation x=\frac{-15±\sqrt{65}}{-1} when ± is plus. Add -15 to \sqrt{65}.

$$x=\frac{\sqrt{65}-15}{-1}$$

Divide -15+\sqrt{65} by -1.

$$x=15-\sqrt{65}$$

Now solve the equation x=\frac{-15±\sqrt{65}}{-1} when ± is minus. Subtract \sqrt{65} from -15.

$$x=\frac{-\sqrt{65}-15}{-1}$$

Divide -15-\sqrt{65} by -1.

$$x=\sqrt{65}+15$$

The equation is now solved.

$$x=15-\sqrt{65} x=\sqrt{65}+15$$

Steps for Completing the Square

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.

$$-\frac{1}{2}x^{2}+15x-80=0$$

Add 80 to both sides of the equation.

$$-\frac{1}{2}x^{2}+15x-80-\left(-80\right)=-\left(-80\right)$$

Subtracting -80 from itself leaves 0.

$$-\frac{1}{2}x^{2}+15x=-\left(-80\right)$$

Subtract -80 from 0.

$$-\frac{1}{2}x^{2}+15x=80$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}x^{2}+15x}{-\frac{1}{2}}=\frac{80}{-\frac{1}{2}}$$

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

$$x^{2}+\frac{15}{-\frac{1}{2}}x=\frac{80}{-\frac{1}{2}}$$

Divide 15 by -\frac{1}{2} by multiplying 15 by the reciprocal of -\frac{1}{2}.

$$x^{2}-30x=\frac{80}{-\frac{1}{2}}$$

Divide 80 by -\frac{1}{2} by multiplying 80 by the reciprocal of -\frac{1}{2}.

$$x^{2}-30x=-160$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-160+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-160+225$$

Add -160 to 225.

$$x^{2}-30x+225=65$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=65$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{65}$$

Simplify.

$$x-15=\sqrt{65} x-15=-\sqrt{65}$$

Add 15 to both sides of the equation.

$$x=\sqrt{65}+15 x=15-\sqrt{65}$$