Solve (10-x)(8+x)=90

$$( 10 - x ) ( 8 + x ) = 90$$

Solve for x (complex solution)

$$x=1-3i$$

$$x=1+3i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10-x by 8+x and combine like terms.

$$80+2x-x^{2}=90$$

Subtract 90 from both sides.

$$80+2x-x^{2}-90=0$$

Subtract 90 from 80 to get -10.

$$-10+2x-x^{2}=0$$

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.

$$-x^{2}+2x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\left(-10\right)}}{2\left(-1\right)}$$

Multiply 4 times -10.

$$x=\frac{-2±\sqrt{4-40}}{2\left(-1\right)}$$

Add 4 to -40.

$$x=\frac{-2±\sqrt{-36}}{2\left(-1\right)}$$

Take the square root of -36.

$$x=\frac{-2±6i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±6i}{-2}$$

Now solve the equation x=\frac{-2±6i}{-2} when ± is plus. Add -2 to 6i.

$$x=\frac{-2+6i}{-2}$$

Divide -2+6i by -2.

$$x=1-3i$$

Now solve the equation x=\frac{-2±6i}{-2} when ± is minus. Subtract 6i from -2.

$$x=\frac{-2-6i}{-2}$$

Divide -2-6i by -2.

$$x=1+3i$$

The equation is now solved.

$$x=1-3i x=1+3i$$

Steps for Completing the Square

Use the distributive property to multiply 10-x by 8+x and combine like terms.

$$80+2x-x^{2}=90$$

Subtract 80 from both sides.

$$2x-x^{2}=90-80$$

Subtract 80 from 90 to get 10.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=\frac{10}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=\frac{10}{-1}$$

Divide 10 by -1.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-10+1$$

Add -10 to 1.

$$x^{2}-2x+1=-9$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-9}$$

Simplify.

$$x-1=3i x-1=-3i$$

Add 1 to both sides of the equation.

$$x=1+3i x=1-3i$$