Solve 6x^2+18x+12

$$6 { x }^{ 2 } +18x+12$$

Factor

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

Solution Steps

Factor out 6.

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

Consider x^{2}+3x+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.

$$a+b=3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

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

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

Factor out x in the first and 2 in the second group.

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

Factor out common term x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

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.

$$6x^{2}+18x+12=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\times 6\times 12}}{2\times 6}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 6\times 12}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-18±\sqrt{324-24\times 12}}{2\times 6}$$

Multiply -24 times 12.

$$x=\frac{-18±\sqrt{324-288}}{2\times 6}$$

Add 324 to -288.

$$x=\frac{-18±\sqrt{36}}{2\times 6}$$

Take the square root of 36.

$$x=\frac{-18±6}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-18±6}{12}$$

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

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

Divide -12 by 12.

$$x=-1$$

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

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

Divide -24 by 12.

$$x=-2$$

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

$$6x^{2}+18x+12=6\left(x-\left(-1\right)\right)\left(x-\left(-2\right)\right)$$

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

$$6x^{2}+18x+12=6\left(x+1\right)\left(x+2\right)$$

Evaluate

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