Solve 100=x*(30-2x)

$$100=x \times (30-2x)$$

Solve for x

$$x=5$$

$$x=10$$

Steps Using the Quadratic Formula

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

$$100=30x-2x^{2}$$

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

$$30x-2x^{2}=100$$

Subtract 100 from both sides.

$$30x-2x^{2}-100=0$$

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.

$$-2x^{2}+30x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 30 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-2\right)\left(-100\right)}}{2\left(-2\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-2\right)\left(-100\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-30±\sqrt{900+8\left(-100\right)}}{2\left(-2\right)}$$

Multiply 8 times -100.

$$x=\frac{-30±\sqrt{900-800}}{2\left(-2\right)}$$

Add 900 to -800.

$$x=\frac{-30±\sqrt{100}}{2\left(-2\right)}$$

Take the square root of 100.

$$x=\frac{-30±10}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-30±10}{-4}$$

Now solve the equation x=\frac{-30±10}{-4} when ± is plus. Add -30 to 10.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-30±10}{-4} when ± is minus. Subtract 10 from -30.

$$x=-\frac{40}{-4}$$

Divide -40 by -4.

$$x=10$$

The equation is now solved.

$$x=5 x=10$$

Steps for Completing the Square

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

$$100=30x-2x^{2}$$

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

$$30x-2x^{2}=100$$

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.

$$-2x^{2}+30x=100$$

Divide both sides by -2.

$$\frac{-2x^{2}+30x}{-2}=\frac{100}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{30}{-2}x=\frac{100}{-2}$$

Divide 30 by -2.

$$x^{2}-15x=\frac{100}{-2}$$

Divide 100 by -2.

$$x^{2}-15x=-50$$

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

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-50+\frac{225}{4}$$

Add -50 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{25}{4}$$

Factor x^{2}-15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{5}{2} x-\frac{15}{2}=-\frac{5}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=10 x=5$$