Solve -2x^2+360x-13000=3000

$$-2 { x }^{ 2 } +360x-13000=3000$$

Solve for x

$$x=80$$

$$x=100$$

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.

$$-2x^{2}+360x-13000=3000$$

Subtract 3000 from both sides of the equation.

$$-2x^{2}+360x-13000-3000=3000-3000$$

Subtracting 3000 from itself leaves 0.

$$-2x^{2}+360x-13000-3000=0$$

Subtract 3000 from -13000.

$$-2x^{2}+360x-16000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 360 for b, and -16000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-360±\sqrt{360^{2}-4\left(-2\right)\left(-16000\right)}}{2\left(-2\right)}$$

Square 360.

$$x=\frac{-360±\sqrt{129600-4\left(-2\right)\left(-16000\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-360±\sqrt{129600+8\left(-16000\right)}}{2\left(-2\right)}$$

Multiply 8 times -16000.

$$x=\frac{-360±\sqrt{129600-128000}}{2\left(-2\right)}$$

Add 129600 to -128000.

$$x=\frac{-360±\sqrt{1600}}{2\left(-2\right)}$$

Take the square root of 1600.

$$x=\frac{-360±40}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-360±40}{-4}$$

Now solve the equation x=\frac{-360±40}{-4} when ± is plus. Add -360 to 40.

$$x=-\frac{320}{-4}$$

Divide -320 by -4.

$$x=80$$

Now solve the equation x=\frac{-360±40}{-4} when ± is minus. Subtract 40 from -360.

$$x=-\frac{400}{-4}$$

Divide -400 by -4.

$$x=100$$

The equation is now solved.

$$x=80 x=100$$

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.

$$-2x^{2}+360x-13000=3000$$

Add 13000 to both sides of the equation.

$$-2x^{2}+360x-13000-\left(-13000\right)=3000-\left(-13000\right)$$

Subtracting -13000 from itself leaves 0.

$$-2x^{2}+360x=3000-\left(-13000\right)$$

Subtract -13000 from 3000.

$$-2x^{2}+360x=16000$$

Divide both sides by -2.

$$\frac{-2x^{2}+360x}{-2}=\frac{16000}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{360}{-2}x=\frac{16000}{-2}$$

Divide 360 by -2.

$$x^{2}-180x=\frac{16000}{-2}$$

Divide 16000 by -2.

$$x^{2}-180x=-8000$$

Divide -180, the coefficient of the x term, by 2 to get -90. Then add the square of -90 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-180x+\left(-90\right)^{2}=-8000+\left(-90\right)^{2}$$

Square -90.

$$x^{2}-180x+8100=-8000+8100$$

Add -8000 to 8100.

$$x^{2}-180x+8100=100$$

Factor x^{2}-180x+8100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-90\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-90\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-90=10 x-90=-10$$

Add 90 to both sides of the equation.

$$x=100 x=80$$