Solve [(150+x)-100](300-10x)=18200

$$[ ( 150 + x ) - 100 ] ( 300 - 10 x ) = 18200$$

Solve for x (complex solution)

$$x=-10+2\sqrt{55}i\approx -10+14.832396974i$$

$$x=-2\sqrt{55}i-10\approx -10-14.832396974i$$

Steps Using the Quadratic Formula

Subtract 100 from 150 to get 50.

$$\left(50+x\right)\left(300-10x\right)=18200$$

Use the distributive property to multiply 50+x by 300-10x and combine like terms.

$$15000-200x-10x^{2}=18200$$

Subtract 18200 from both sides.

$$15000-200x-10x^{2}-18200=0$$

Subtract 18200 from 15000 to get -3200.

$$-3200-200x-10x^{2}=0$$

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.

$$-10x^{2}-200x-3200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, -200 for b, and -3200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\left(-10\right)\left(-3200\right)}}{2\left(-10\right)}$$

Square -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\left(-10\right)\left(-3200\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-\left(-200\right)±\sqrt{40000+40\left(-3200\right)}}{2\left(-10\right)}$$

Multiply 40 times -3200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-128000}}{2\left(-10\right)}$$

Add 40000 to -128000.

$$x=\frac{-\left(-200\right)±\sqrt{-88000}}{2\left(-10\right)}$$

Take the square root of -88000.

$$x=\frac{-\left(-200\right)±40\sqrt{55}i}{2\left(-10\right)}$$

The opposite of -200 is 200.

$$x=\frac{200±40\sqrt{55}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{200±40\sqrt{55}i}{-20}$$

Now solve the equation x=\frac{200±40\sqrt{55}i}{-20} when ± is plus. Add 200 to 40i\sqrt{55}.

$$x=\frac{200+40\sqrt{55}i}{-20}$$

Divide 200+40i\sqrt{55} by -20.

$$x=-2\sqrt{55}i-10$$

Now solve the equation x=\frac{200±40\sqrt{55}i}{-20} when ± is minus. Subtract 40i\sqrt{55} from 200.

$$x=\frac{-40\sqrt{55}i+200}{-20}$$

Divide 200-40i\sqrt{55} by -20.

$$x=-10+2\sqrt{55}i$$

The equation is now solved.

$$x=-2\sqrt{55}i-10 x=-10+2\sqrt{55}i$$

Steps for Completing the Square

Subtract 100 from 150 to get 50.

$$\left(50+x\right)\left(300-10x\right)=18200$$

Use the distributive property to multiply 50+x by 300-10x and combine like terms.

$$15000-200x-10x^{2}=18200$$

Subtract 15000 from both sides.

$$-200x-10x^{2}=18200-15000$$

Subtract 15000 from 18200 to get 3200.

$$-200x-10x^{2}=3200$$

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.

$$-10x^{2}-200x=3200$$

Divide both sides by -10.

$$\frac{-10x^{2}-200x}{-10}=\frac{3200}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\left(-\frac{200}{-10}\right)x=\frac{3200}{-10}$$

Divide -200 by -10.

$$x^{2}+20x=\frac{3200}{-10}$$

Divide 3200 by -10.

$$x^{2}+20x=-320$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=-320+10^{2}$$

Square 10.

$$x^{2}+20x+100=-320+100$$

Add -320 to 100.

$$x^{2}+20x+100=-220$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=-220$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{-220}$$

Simplify.

$$x+10=2\sqrt{55}i x+10=-2\sqrt{55}i$$

Subtract 10 from both sides of the equation.

$$x=-10+2\sqrt{55}i x=-2\sqrt{55}i-10$$