Solve (12-3x)(6-2x)=36

$$( 12 - 3 x ) ( 6 - 2 x ) = 36$$

Solve for x

$$x=1$$

$$x=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12-3x by 6-2x and combine like terms.

$$72-42x+6x^{2}=36$$

Subtract 36 from both sides.

$$72-42x+6x^{2}-36=0$$

Subtract 36 from 72 to get 36.

$$36-42x+6x^{2}=0$$

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.

$$6x^{2}-42x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -42 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 6\times 36}}{2\times 6}$$

Square -42.

$$x=\frac{-\left(-42\right)±\sqrt{1764-4\times 6\times 36}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-42\right)±\sqrt{1764-24\times 36}}{2\times 6}$$

Multiply -24 times 36.

$$x=\frac{-\left(-42\right)±\sqrt{1764-864}}{2\times 6}$$

Add 1764 to -864.

$$x=\frac{-\left(-42\right)±\sqrt{900}}{2\times 6}$$

Take the square root of 900.

$$x=\frac{-\left(-42\right)±30}{2\times 6}$$

The opposite of -42 is 42.

$$x=\frac{42±30}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{42±30}{12}$$

Now solve the equation x=\frac{42±30}{12} when ± is plus. Add 42 to 30.

$$x=\frac{72}{12}$$

Divide 72 by 12.

$$x=6$$

Now solve the equation x=\frac{42±30}{12} when ± is minus. Subtract 30 from 42.

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

Divide 12 by 12.

$$x=1$$

The equation is now solved.

$$x=6 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 12-3x by 6-2x and combine like terms.

$$72-42x+6x^{2}=36$$

Subtract 72 from both sides.

$$-42x+6x^{2}=36-72$$

Subtract 72 from 36 to get -36.

$$-42x+6x^{2}=-36$$

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.

$$6x^{2}-42x=-36$$

Divide both sides by 6.

$$\frac{6x^{2}-42x}{6}=-\frac{36}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{42}{6}\right)x=-\frac{36}{6}$$

Divide -42 by 6.

$$x^{2}-7x=-\frac{36}{6}$$

Divide -36 by 6.

$$x^{2}-7x=-6$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-6+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-6+\frac{49}{4}$$

Add -6 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{25}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{5}{2} x-\frac{7}{2}=-\frac{5}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=6 x=1$$