Solve x^2+19x+84

$${ x }^{ 2 } +19x+84$$

Factor

$$\left(x+7\right)\left(x+12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+84. To find a and b, set up a system to be solved.

$$a+b=19 ab=1\times 84=84$$

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 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 19.

$$a=7 b=12$$

Rewrite x^{2}+19x+84 as \left(x^{2}+7x\right)+\left(12x+84\right).

$$\left(x^{2}+7x\right)+\left(12x+84\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x+7\right)+12\left(x+7\right)$$

Factor out common term x+7 by using distributive property.

$$\left(x+7\right)\left(x+12\right)$$

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.

$$x^{2}+19x+84=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\times 84}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 84}}{2}$$

Multiply -4 times 84.

$$x=\frac{-19±\sqrt{361-336}}{2}$$

Add 361 to -336.

$$x=\frac{-19±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-19±5}{2}$$

Now solve the equation x=\frac{-19±5}{2} when ± is plus. Add -19 to 5.

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

Divide -14 by 2.

$$x=-7$$

Now solve the equation x=\frac{-19±5}{2} when ± is minus. Subtract 5 from -19.

$$x=-\frac{24}{2}$$

Divide -24 by 2.

$$x=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and -12 for x_{2}.

$$x^{2}+19x+84=\left(x-\left(-7\right)\right)\left(x-\left(-12\right)\right)$$

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

$$x^{2}+19x+84=\left(x+7\right)\left(x+12\right)$$

Evaluate

$$\left(x+7\right)\left(x+12\right)$$