Solve 1500=(40+x)(20+x)

$$1500=(40+x)(20+x)$$

Solve for x

$$x=-70$$

$$x=10$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40+x by 20+x and combine like terms.

$$1500=800+60x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$800+60x+x^{2}=1500$$

Subtract 1500 from both sides.

$$800+60x+x^{2}-1500=0$$

Subtract 1500 from 800 to get -700.

$$-700+60x+x^{2}=0$$

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.

$$x^{2}+60x-700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 60 for b, and -700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-700\right)}}{2}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-700\right)}}{2}$$

Multiply -4 times -700.

$$x=\frac{-60±\sqrt{3600+2800}}{2}$$

Add 3600 to 2800.

$$x=\frac{-60±\sqrt{6400}}{2}$$

Take the square root of 6400.

$$x=\frac{-60±80}{2}$$

Now solve the equation x=\frac{-60±80}{2} when ± is plus. Add -60 to 80.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{-60±80}{2} when ± is minus. Subtract 80 from -60.

$$x=-\frac{140}{2}$$

Divide -140 by 2.

$$x=-70$$

The equation is now solved.

$$x=10 x=-70$$

Steps for Completing the Square

Use the distributive property to multiply 40+x by 20+x and combine like terms.

$$1500=800+60x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$800+60x+x^{2}=1500$$

Subtract 800 from both sides.

$$60x+x^{2}=1500-800$$

Subtract 800 from 1500 to get 700.

$$60x+x^{2}=700$$

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.

$$x^{2}+60x=700$$

Divide 60, the coefficient of the x term, by 2 to get 30. Then add the square of 30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+60x+30^{2}=700+30^{2}$$

Square 30.

$$x^{2}+60x+900=700+900$$

Add 700 to 900.

$$x^{2}+60x+900=1600$$

Factor x^{2}+60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+30\right)^{2}=1600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{1600}$$

Simplify.

$$x+30=40 x+30=-40$$

Subtract 30 from both sides of the equation.

$$x=10 x=-70$$