Solve x*(30-2x)=72

$$x \times (30-2x)=72$$

Solve for x

$$x=3$$

$$x=12$$

Steps Using the Quadratic Formula

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

$$30x-2x^{2}=72$$

Subtract 72 from both sides.

$$30x-2x^{2}-72=0$$

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.

$$-2x^{2}+30x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 30 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-2\right)\left(-72\right)}}{2\left(-2\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-2\right)\left(-72\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-30±\sqrt{900+8\left(-72\right)}}{2\left(-2\right)}$$

Multiply 8 times -72.

$$x=\frac{-30±\sqrt{900-576}}{2\left(-2\right)}$$

Add 900 to -576.

$$x=\frac{-30±\sqrt{324}}{2\left(-2\right)}$$

Take the square root of 324.

$$x=\frac{-30±18}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-30±18}{-4}$$

Now solve the equation x=\frac{-30±18}{-4} when ± is plus. Add -30 to 18.

$$x=-\frac{12}{-4}$$

Divide -12 by -4.

$$x=3$$

Now solve the equation x=\frac{-30±18}{-4} when ± is minus. Subtract 18 from -30.

$$x=-\frac{48}{-4}$$

Divide -48 by -4.

$$x=12$$

The equation is now solved.

$$x=3 x=12$$

Steps for Completing the Square

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

$$30x-2x^{2}=72$$

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.

$$-2x^{2}+30x=72$$

Divide both sides by -2.

$$\frac{-2x^{2}+30x}{-2}=\frac{72}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{30}{-2}x=\frac{72}{-2}$$

Divide 30 by -2.

$$x^{2}-15x=\frac{72}{-2}$$

Divide 72 by -2.

$$x^{2}-15x=-36$$

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.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-36+\left(-\frac{15}{2}\right)^{2}$$

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

$$x^{2}-15x+\frac{225}{4}=-36+\frac{225}{4}$$

Add -36 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{81}{4}$$

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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{9}{2} x-\frac{15}{2}=-\frac{9}{2}$$

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

$$x=12 x=3$$