Solve x^2-19x+84

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

Factor

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

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 negative, a and b are both negative. 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.

$$a=-12 b=-7$$

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

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

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

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

Factor out common term x-12 by using distributive property.

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

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.

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

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.

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

Square -19.

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

Multiply -4 times 84.

$$x=\frac{-\left(-19\right)±\sqrt{361-336}}{2}$$

Add 361 to -336.

$$x=\frac{-\left(-19\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-19\right)±5}{2}$$

The opposite of -19 is 19.

$$x=\frac{19±5}{2}$$

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

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

Divide 24 by 2.

$$x=12$$

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

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

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 12 for x_{1} and 7 for x_{2}.

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

Evaluate

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