Solve (1000-5x)(x-100)=32000

$$(1000-5x)(x-100)=32000$$

Solve for x (complex solution)

$$x=150+10\sqrt{39}i\approx 150+62.449979984i$$

$$x=-10\sqrt{39}i+150\approx 150-62.449979984i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1000-5x by x-100 and combine like terms.

$$1500x-100000-5x^{2}=32000$$

Subtract 32000 from both sides.

$$1500x-100000-5x^{2}-32000=0$$

Subtract 32000 from -100000 to get -132000.

$$1500x-132000-5x^{2}=0$$

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.

$$-5x^{2}+1500x-132000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 1500 for b, and -132000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1500±\sqrt{1500^{2}-4\left(-5\right)\left(-132000\right)}}{2\left(-5\right)}$$

Square 1500.

$$x=\frac{-1500±\sqrt{2250000-4\left(-5\right)\left(-132000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-1500±\sqrt{2250000+20\left(-132000\right)}}{2\left(-5\right)}$$

Multiply 20 times -132000.

$$x=\frac{-1500±\sqrt{2250000-2640000}}{2\left(-5\right)}$$

Add 2250000 to -2640000.

$$x=\frac{-1500±\sqrt{-390000}}{2\left(-5\right)}$$

Take the square root of -390000.

$$x=\frac{-1500±100\sqrt{39}i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-1500±100\sqrt{39}i}{-10}$$

Now solve the equation x=\frac{-1500±100\sqrt{39}i}{-10} when ± is plus. Add -1500 to 100i\sqrt{39}.

$$x=\frac{-1500+100\sqrt{39}i}{-10}$$

Divide -1500+100i\sqrt{39} by -10.

$$x=-10\sqrt{39}i+150$$

Now solve the equation x=\frac{-1500±100\sqrt{39}i}{-10} when ± is minus. Subtract 100i\sqrt{39} from -1500.

$$x=\frac{-100\sqrt{39}i-1500}{-10}$$

Divide -1500-100i\sqrt{39} by -10.

$$x=150+10\sqrt{39}i$$

The equation is now solved.

$$x=-10\sqrt{39}i+150 x=150+10\sqrt{39}i$$

Steps for Completing the Square

Use the distributive property to multiply 1000-5x by x-100 and combine like terms.

$$1500x-100000-5x^{2}=32000$$

Add 100000 to both sides.

$$1500x-5x^{2}=32000+100000$$

Add 32000 and 100000 to get 132000.

$$1500x-5x^{2}=132000$$

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.

$$-5x^{2}+1500x=132000$$

Divide both sides by -5.

$$\frac{-5x^{2}+1500x}{-5}=\frac{132000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{1500}{-5}x=\frac{132000}{-5}$$

Divide 1500 by -5.

$$x^{2}-300x=\frac{132000}{-5}$$

Divide 132000 by -5.

$$x^{2}-300x=-26400$$

Divide -300, the coefficient of the x term, by 2 to get -150. Then add the square of -150 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-300x+\left(-150\right)^{2}=-26400+\left(-150\right)^{2}$$

Square -150.

$$x^{2}-300x+22500=-26400+22500$$

Add -26400 to 22500.

$$x^{2}-300x+22500=-3900$$

Factor x^{2}-300x+22500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-150\right)^{2}=-3900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-150\right)^{2}}=\sqrt{-3900}$$

Simplify.

$$x-150=10\sqrt{39}i x-150=-10\sqrt{39}i$$

Add 150 to both sides of the equation.

$$x=150+10\sqrt{39}i x=-10\sqrt{39}i+150$$