Solve x^2+11x+28

$${ x }^{ 2 } +11x+28$$

Factor

$$\left(x+4\right)\left(x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 28=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=4 b=7$$

Rewrite x^{2}+11x+28 as \left(x^{2}+4x\right)+\left(7x+28\right).

$$\left(x^{2}+4x\right)+\left(7x+28\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x+4\right)+7\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+7\right)$$

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.

$$x^{2}+11x+28=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 28}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 28}}{2}$$

Multiply -4 times 28.

$$x=\frac{-11±\sqrt{121-112}}{2}$$

Add 121 to -112.

$$x=\frac{-11±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-11±3}{2}$$

Now solve the equation x=\frac{-11±3}{2} when ± is plus. Add -11 to 3.

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

Divide -8 by 2.

$$x=-4$$

Now solve the equation x=\frac{-11±3}{2} when ± is minus. Subtract 3 from -11.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4 for x_{1} and -7 for x_{2}.

$$x^{2}+11x+28=\left(x-\left(-4\right)\right)\left(x-\left(-7\right)\right)$$

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

$$x^{2}+11x+28=\left(x+4\right)\left(x+7\right)$$

Evaluate

$$\left(x+4\right)\left(x+7\right)$$