Solve 400=x(40-x)

$$400=x(40-x)$$

Solve for x

$$x=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 40-x.

$$400=40x-x^{2}$$

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

$$40x-x^{2}=400$$

Subtract 400 from both sides.

$$40x-x^{2}-400=0$$

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.

$$-x^{2}+40x-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 40 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-1\right)\left(-400\right)}}{2\left(-1\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-1\right)\left(-400\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-40±\sqrt{1600+4\left(-400\right)}}{2\left(-1\right)}$$

Multiply 4 times -400.

$$x=\frac{-40±\sqrt{1600-1600}}{2\left(-1\right)}$$

Add 1600 to -1600.

$$x=\frac{-40±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{40}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{40}{-2}$$

Divide -40 by -2.

$$x=20$$

Steps for Completing the Square

Use the distributive property to multiply x by 40-x.

$$400=40x-x^{2}$$

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

$$40x-x^{2}=400$$

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.

$$-x^{2}+40x=400$$

Divide both sides by -1.

$$\frac{-x^{2}+40x}{-1}=\frac{400}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{40}{-1}x=\frac{400}{-1}$$

Divide 40 by -1.

$$x^{2}-40x=\frac{400}{-1}$$

Divide 400 by -1.

$$x^{2}-40x=-400$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-400+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-400+400$$

Add -400 to 400.

$$x^{2}-40x+400=0$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-20=0 x-20=0$$

Add 20 to both sides of the equation.

$$x=20 x=20$$

The equation is now solved. Solutions are the same.

$$x=20$$