Solve x^2+6x+8+0

$${ x }^{ 2 } +6x+8+0$$

Factor

$$\left(x+2\right)\left(x+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\times 8=8$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 6.

$$a=2 b=4$$

Rewrite x^{2}+6x+8 as \left(x^{2}+2x\right)+\left(4x+8\right).

$$\left(x^{2}+2x\right)+\left(4x+8\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x+2\right)+4\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+4\right)$$

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.

$$x^{2}+6x+8=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 8}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-6±\sqrt{36-32}}{2}$$

Add 36 to -32.

$$x=\frac{-6±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-6±2}{2}$$

Now solve the equation x=\frac{-6±2}{2} when ± is plus. Add -6 to 2.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-6±2}{2} when ± is minus. Subtract 2 from -6.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

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 -4 for x_{2}.

$$x^{2}+6x+8=\left(x-\left(-2\right)\right)\left(x-\left(-4\right)\right)$$

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

$$x^{2}+6x+8=\left(x+2\right)\left(x+4\right)$$

Evaluate

$$\left(x+2\right)\left(x+4\right)$$

Solution Steps

Add 8 and 0 to get 8.

$$x^{2}+6x+8$$