Solve (30-3x)(10-2x)=216

$$(30-3x)(10-2x)=216$$

Solve for x

$$x=1$$

$$x=14$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 30-3x by 10-2x and combine like terms.

$$300-90x+6x^{2}=216$$

Subtract 216 from both sides.

$$300-90x+6x^{2}-216=0$$

Subtract 216 from 300 to get 84.

$$84-90x+6x^{2}=0$$

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.

$$6x^{2}-90x+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -90 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 6\times 84}}{2\times 6}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 6\times 84}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-90\right)±\sqrt{8100-24\times 84}}{2\times 6}$$

Multiply -24 times 84.

$$x=\frac{-\left(-90\right)±\sqrt{8100-2016}}{2\times 6}$$

Add 8100 to -2016.

$$x=\frac{-\left(-90\right)±\sqrt{6084}}{2\times 6}$$

Take the square root of 6084.

$$x=\frac{-\left(-90\right)±78}{2\times 6}$$

The opposite of -90 is 90.

$$x=\frac{90±78}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{90±78}{12}$$

Now solve the equation x=\frac{90±78}{12} when ± is plus. Add 90 to 78.

$$x=\frac{168}{12}$$

Divide 168 by 12.

$$x=14$$

Now solve the equation x=\frac{90±78}{12} when ± is minus. Subtract 78 from 90.

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

Divide 12 by 12.

$$x=1$$

The equation is now solved.

$$x=14 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 30-3x by 10-2x and combine like terms.

$$300-90x+6x^{2}=216$$

Subtract 300 from both sides.

$$-90x+6x^{2}=216-300$$

Subtract 300 from 216 to get -84.

$$-90x+6x^{2}=-84$$

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.

$$6x^{2}-90x=-84$$

Divide both sides by 6.

$$\frac{6x^{2}-90x}{6}=-\frac{84}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{90}{6}\right)x=-\frac{84}{6}$$

Divide -90 by 6.

$$x^{2}-15x=-\frac{84}{6}$$

Divide -84 by 6.

$$x^{2}-15x=-14$$

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.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-14+\left(-\frac{15}{2}\right)^{2}$$

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

$$x^{2}-15x+\frac{225}{4}=-14+\frac{225}{4}$$

Add -14 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{169}{4}$$

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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{13}{2} x-\frac{15}{2}=-\frac{13}{2}$$

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

$$x=14 x=1$$