Solve 2(x+3)(x-6)/x=0

$$\frac{ 2(x+3)(x-6) }{ x } =0$$

Solve for x

$$x=6$$

$$x=-3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$2\left(x+3\right)\left(x-6\right)=0$$

Use the distributive property to multiply 2 by x+3.

$$\left(2x+6\right)\left(x-6\right)=0$$

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

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

Divide both sides by 2.

$$x^{2}-3x-18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\left(-18\right)=-18$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -3.

$$a=-6 b=3$$

Rewrite x^{2}-3x-18 as \left(x^{2}-6x\right)+\left(3x-18\right).

$$\left(x^{2}-6x\right)+\left(3x-18\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-6\right)+3\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+3\right)$$

To find equation solutions, solve x-6=0 and x+3=0.

$$x=6 x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$2\left(x+3\right)\left(x-6\right)=0$$

Use the distributive property to multiply 2 by x+3.

$$\left(2x+6\right)\left(x-6\right)=0$$

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

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

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

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

Square -6.

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

Multiply -4 times 2.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\left(-36\right)}}{2\times 2}$$

Multiply -8 times -36.

$$x=\frac{-\left(-6\right)±\sqrt{36+288}}{2\times 2}$$

Add 36 to 288.

$$x=\frac{-\left(-6\right)±\sqrt{324}}{2\times 2}$$

Take the square root of 324.

$$x=\frac{-\left(-6\right)±18}{2\times 2}$$

The opposite of -6 is 6.

$$x=\frac{6±18}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{6±18}{4}$$

Now solve the equation x=\frac{6±18}{4} when ± is plus. Add 6 to 18.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

Now solve the equation x=\frac{6±18}{4} when ± is minus. Subtract 18 from 6.

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

Divide -12 by 4.

$$x=-3$$

The equation is now solved.

$$x=6 x=-3$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$2\left(x+3\right)\left(x-6\right)=0$$

Use the distributive property to multiply 2 by x+3.

$$\left(2x+6\right)\left(x-6\right)=0$$

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

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

Add 36 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -6 by 2.

$$x^{2}-3x=\frac{36}{2}$$

Divide 36 by 2.

$$x^{2}-3x=18$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=18+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=18+\frac{9}{4}$$

Add 18 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{81}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{9}{2} x-\frac{3}{2}=-\frac{9}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=6 x=-3$$