Solve 2x(30-2x)+2x(20-2x)=200

$$2x(30-2x)+2x(20-2x)=200$$

Solve for x

$$x=10$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 30-2x.

$$60x-4x^{2}+2x\left(20-2x\right)=200$$

Use the distributive property to multiply 2x by 20-2x.

$$60x-4x^{2}+40x-4x^{2}=200$$

Combine 60x and 40x to get 100x.

$$100x-4x^{2}-4x^{2}=200$$

Combine -4x^{2} and -4x^{2} to get -8x^{2}.

$$100x-8x^{2}=200$$

Subtract 200 from both sides.

$$100x-8x^{2}-200=0$$

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.

$$-8x^{2}+100x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 100 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-8\right)\left(-200\right)}}{2\left(-8\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-8\right)\left(-200\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-100±\sqrt{10000+32\left(-200\right)}}{2\left(-8\right)}$$

Multiply 32 times -200.

$$x=\frac{-100±\sqrt{10000-6400}}{2\left(-8\right)}$$

Add 10000 to -6400.

$$x=\frac{-100±\sqrt{3600}}{2\left(-8\right)}$$

Take the square root of 3600.

$$x=\frac{-100±60}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-100±60}{-16}$$

Now solve the equation x=\frac{-100±60}{-16} when ± is plus. Add -100 to 60.

$$x=-\frac{40}{-16}$$

Reduce the fraction \frac{-40}{-16} to lowest terms by extracting and canceling out 8.

$$x=\frac{5}{2}$$

Now solve the equation x=\frac{-100±60}{-16} when ± is minus. Subtract 60 from -100.

$$x=-\frac{160}{-16}$$

Divide -160 by -16.

$$x=10$$

The equation is now solved.

$$x=\frac{5}{2} x=10$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 30-2x.

$$60x-4x^{2}+2x\left(20-2x\right)=200$$

Use the distributive property to multiply 2x by 20-2x.

$$60x-4x^{2}+40x-4x^{2}=200$$

Combine 60x and 40x to get 100x.

$$100x-4x^{2}-4x^{2}=200$$

Combine -4x^{2} and -4x^{2} to get -8x^{2}.

$$100x-8x^{2}=200$$

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.

$$-8x^{2}+100x=200$$

Divide both sides by -8.

$$\frac{-8x^{2}+100x}{-8}=\frac{200}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{100}{-8}x=\frac{200}{-8}$$

Reduce the fraction \frac{100}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{25}{2}x=\frac{200}{-8}$$

Divide 200 by -8.

$$x^{2}-\frac{25}{2}x=-25$$

Divide -\frac{25}{2}, the coefficient of the x term, by 2 to get -\frac{25}{4}. Then add the square of -\frac{25}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{2}x+\left(-\frac{25}{4}\right)^{2}=-25+\left(-\frac{25}{4}\right)^{2}$$

Square -\frac{25}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{2}x+\frac{625}{16}=-25+\frac{625}{16}$$

Add -25 to \frac{625}{16}.

$$x^{2}-\frac{25}{2}x+\frac{625}{16}=\frac{225}{16}$$

Factor x^{2}-\frac{25}{2}x+\frac{625}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x-\frac{25}{4}=\frac{15}{4} x-\frac{25}{4}=-\frac{15}{4}$$

Add \frac{25}{4} to both sides of the equation.

$$x=10 x=\frac{5}{2}$$