Solve 6x^2-6x-12

$$6 { x }^{ 2 } -6x-12$$

Factor

$$6\left(x-2\right)\left(x+1\right)$$

Solution Steps

Factor out 6.

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

Consider x^{2}-x-2. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-2\right)=-2$$

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. The only such pair is the system solution.

$$a=-2 b=1$$

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

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

Factor out x in x^{2}-2x.

$$x\left(x-2\right)+x-2$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$6\left(x-2\right)\left(x+1\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$6x^{2}-6x-12=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 6\left(-12\right)}}{2\times 6}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 6\left(-12\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-6\right)±\sqrt{36-24\left(-12\right)}}{2\times 6}$$

Multiply -24 times -12.

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

Add 36 to 288.

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

Take the square root of 324.

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

The opposite of -6 is 6.

$$x=\frac{6±18}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{6±18}{12}$$

Now solve the equation x=\frac{6±18}{12} when ± is plus. Add 6 to 18.

$$x=\frac{24}{12}$$

Divide 24 by 12.

$$x=2$$

Now solve the equation x=\frac{6±18}{12} when ± is minus. Subtract 18 from 6.

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

Divide -12 by 12.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -1 for x_{2}.

$$6x^{2}-6x-12=6\left(x-2\right)\left(x-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$6x^{2}-6x-12=6\left(x-2\right)\left(x+1\right)$$

Evaluate

$$6\left(x-2\right)\left(x+1\right)$$