Solve (10+x)(400-10x)=6000

$$(10+x)(400-10x)=6000$$

Solve for x

$$x=10$$

$$x=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+x by 400-10x and combine like terms.

$$4000+300x-10x^{2}=6000$$

Subtract 6000 from both sides.

$$4000+300x-10x^{2}-6000=0$$

Subtract 6000 from 4000 to get -2000.

$$-2000+300x-10x^{2}=0$$

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.

$$-10x^{2}+300x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 300 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-10\right)\left(-2000\right)}}{2\left(-10\right)}$$

Square 300.

$$x=\frac{-300±\sqrt{90000-4\left(-10\right)\left(-2000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-300±\sqrt{90000+40\left(-2000\right)}}{2\left(-10\right)}$$

Multiply 40 times -2000.

$$x=\frac{-300±\sqrt{90000-80000}}{2\left(-10\right)}$$

Add 90000 to -80000.

$$x=\frac{-300±\sqrt{10000}}{2\left(-10\right)}$$

Take the square root of 10000.

$$x=\frac{-300±100}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-300±100}{-20}$$

Now solve the equation x=\frac{-300±100}{-20} when ± is plus. Add -300 to 100.

$$x=-\frac{200}{-20}$$

Divide -200 by -20.

$$x=10$$

Now solve the equation x=\frac{-300±100}{-20} when ± is minus. Subtract 100 from -300.

$$x=-\frac{400}{-20}$$

Divide -400 by -20.

$$x=20$$

The equation is now solved.

$$x=10 x=20$$

Steps for Completing the Square

Use the distributive property to multiply 10+x by 400-10x and combine like terms.

$$4000+300x-10x^{2}=6000$$

Subtract 4000 from both sides.

$$300x-10x^{2}=6000-4000$$

Subtract 4000 from 6000 to get 2000.

$$300x-10x^{2}=2000$$

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.

$$-10x^{2}+300x=2000$$

Divide both sides by -10.

$$\frac{-10x^{2}+300x}{-10}=\frac{2000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{300}{-10}x=\frac{2000}{-10}$$

Divide 300 by -10.

$$x^{2}-30x=\frac{2000}{-10}$$

Divide 2000 by -10.

$$x^{2}-30x=-200$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-200+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-200+225$$

Add -200 to 225.

$$x^{2}-30x+225=25$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-15=5 x-15=-5$$

Add 15 to both sides of the equation.

$$x=20 x=10$$