Solve x*(150-x)=3600

$$x \times (150-x)=3600$$

Solve for x

$$x=30$$

$$x=120$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 150-x.

$$150x-x^{2}=3600$$

Subtract 3600 from both sides.

$$150x-x^{2}-3600=0$$

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.

$$-x^{2}+150x-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 150 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\left(-1\right)\left(-3600\right)}}{2\left(-1\right)}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\left(-1\right)\left(-3600\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-150±\sqrt{22500+4\left(-3600\right)}}{2\left(-1\right)}$$

Multiply 4 times -3600.

$$x=\frac{-150±\sqrt{22500-14400}}{2\left(-1\right)}$$

Add 22500 to -14400.

$$x=\frac{-150±\sqrt{8100}}{2\left(-1\right)}$$

Take the square root of 8100.

$$x=\frac{-150±90}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-150±90}{-2}$$

Now solve the equation x=\frac{-150±90}{-2} when ± is plus. Add -150 to 90.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

Now solve the equation x=\frac{-150±90}{-2} when ± is minus. Subtract 90 from -150.

$$x=-\frac{240}{-2}$$

Divide -240 by -2.

$$x=120$$

The equation is now solved.

$$x=30 x=120$$

Steps for Completing the Square

Use the distributive property to multiply x by 150-x.

$$150x-x^{2}=3600$$

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.

$$-x^{2}+150x=3600$$

Divide both sides by -1.

$$\frac{-x^{2}+150x}{-1}=\frac{3600}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{150}{-1}x=\frac{3600}{-1}$$

Divide 150 by -1.

$$x^{2}-150x=\frac{3600}{-1}$$

Divide 3600 by -1.

$$x^{2}-150x=-3600$$

Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-150x+\left(-75\right)^{2}=-3600+\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=-3600+5625$$

Add -3600 to 5625.

$$x^{2}-150x+5625=2025$$

Factor x^{2}-150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-75\right)^{2}=2025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{2025}$$

Simplify.

$$x-75=45 x-75=-45$$

Add 75 to both sides of the equation.

$$x=120 x=30$$