Solve (100+2x)(60+2x)=200*60

$$( 100 + 2 x ) ( 60 + 2 x ) = 200 \times 60$$

Solve for x

$$x=10\sqrt{31}-40\approx 15.677643628$$

$$x=-10\sqrt{31}-40\approx -95.677643628$$

Steps Using the Quadratic Formula

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

$$6000+320x+4x^{2}=200\times 60$$

Multiply 200 and 60 to get 12000.

$$6000+320x+4x^{2}=12000$$

Subtract 12000 from both sides.

$$6000+320x+4x^{2}-12000=0$$

Subtract 12000 from 6000 to get -6000.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 320 for b, and -6000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-320±\sqrt{320^{2}-4\times 4\left(-6000\right)}}{2\times 4}$$

Square 320.

$$x=\frac{-320±\sqrt{102400-4\times 4\left(-6000\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-320±\sqrt{102400-16\left(-6000\right)}}{2\times 4}$$

Multiply -16 times -6000.

$$x=\frac{-320±\sqrt{102400+96000}}{2\times 4}$$

Add 102400 to 96000.

$$x=\frac{-320±\sqrt{198400}}{2\times 4}$$

Take the square root of 198400.

$$x=\frac{-320±80\sqrt{31}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-320±80\sqrt{31}}{8}$$

Now solve the equation x=\frac{-320±80\sqrt{31}}{8} when ± is plus. Add -320 to 80\sqrt{31}.

$$x=\frac{80\sqrt{31}-320}{8}$$

Divide -320+80\sqrt{31} by 8.

$$x=10\sqrt{31}-40$$

Now solve the equation x=\frac{-320±80\sqrt{31}}{8} when ± is minus. Subtract 80\sqrt{31} from -320.

$$x=\frac{-80\sqrt{31}-320}{8}$$

Divide -320-80\sqrt{31} by 8.

$$x=-10\sqrt{31}-40$$

The equation is now solved.

$$x=10\sqrt{31}-40 x=-10\sqrt{31}-40$$

Steps for Completing the Square

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

$$6000+320x+4x^{2}=200\times 60$$

Multiply 200 and 60 to get 12000.

$$6000+320x+4x^{2}=12000$$

Subtract 6000 from both sides.

$$320x+4x^{2}=12000-6000$$

Subtract 6000 from 12000 to get 6000.

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

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.

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

Divide both sides by 4.

$$\frac{4x^{2}+320x}{4}=\frac{6000}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{320}{4}x=\frac{6000}{4}$$

Divide 320 by 4.

$$x^{2}+80x=\frac{6000}{4}$$

Divide 6000 by 4.

$$x^{2}+80x=1500$$

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.

$$x^{2}+80x+40^{2}=1500+40^{2}$$

Square 40.

$$x^{2}+80x+1600=1500+1600$$

Add 1500 to 1600.

$$x^{2}+80x+1600=3100$$

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}.

$$\left(x+40\right)^{2}=3100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{3100}$$

Simplify.

$$x+40=10\sqrt{31} x+40=-10\sqrt{31}$$

Subtract 40 from both sides of the equation.

$$x=10\sqrt{31}-40 x=-10\sqrt{31}-40$$