Solve (100+2x)(60+2x)=1200

$$(100+2x)(60+2x)=1200$$

Solve for x

$$x=-60$$

$$x=-20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100+2x by 60+2x and combine like terms.

$$6000+320x+4x^{2}=1200$$

Subtract 1200 from both sides.

$$6000+320x+4x^{2}-1200=0$$

Subtract 1200 from 6000 to get 4800.

$$4800+320x+4x^{2}=0$$

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.

$$4x^{2}+320x+4800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 320 for b, and 4800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-320±\sqrt{320^{2}-4\times 4\times 4800}}{2\times 4}$$

Square 320.

$$x=\frac{-320±\sqrt{102400-4\times 4\times 4800}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-320±\sqrt{102400-16\times 4800}}{2\times 4}$$

Multiply -16 times 4800.

$$x=\frac{-320±\sqrt{102400-76800}}{2\times 4}$$

Add 102400 to -76800.

$$x=\frac{-320±\sqrt{25600}}{2\times 4}$$

Take the square root of 25600.

$$x=\frac{-320±160}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-320±160}{8}$$

Now solve the equation x=\frac{-320±160}{8} when ± is plus. Add -320 to 160.

$$x=-\frac{160}{8}$$

Divide -160 by 8.

$$x=-20$$

Now solve the equation x=\frac{-320±160}{8} when ± is minus. Subtract 160 from -320.

$$x=-\frac{480}{8}$$

Divide -480 by 8.

$$x=-60$$

The equation is now solved.

$$x=-20 x=-60$$

Steps for Completing the Square

Use the distributive property to multiply 100+2x by 60+2x and combine like terms.

$$6000+320x+4x^{2}=1200$$

Subtract 6000 from both sides.

$$320x+4x^{2}=1200-6000$$

Subtract 6000 from 1200 to get -4800.

$$320x+4x^{2}=-4800$$

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.

$$4x^{2}+320x=-4800$$

Divide both sides by 4.

$$\frac{4x^{2}+320x}{4}=-\frac{4800}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{320}{4}x=-\frac{4800}{4}$$

Divide 320 by 4.

$$x^{2}+80x=-\frac{4800}{4}$$

Divide -4800 by 4.

$$x^{2}+80x=-1200$$

Divide 80, the coefficient of the x term, by 2 to get 40. Then add the square of 40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+80x+40^{2}=-1200+40^{2}$$

Square 40.

$$x^{2}+80x+1600=-1200+1600$$

Add -1200 to 1600.

$$x^{2}+80x+1600=400$$

Factor x^{2}+80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+40\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{400}$$

Simplify.

$$x+40=20 x+40=-20$$

Subtract 40 from both sides of the equation.

$$x=-20 x=-60$$