Solve x^2-17x+66

$${ x }^{ 2 } -17x+66$$

Factor

$$\left(x-11\right)\left(x-6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+66. To find a and b, set up a system to be solved.

$$a+b=-17 ab=1\times 66=66$$

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 66.

$$-1,-66 -2,-33 -3,-22 -6,-11$$

Calculate the sum for each pair.

$$-1-66=-67 -2-33=-35 -3-22=-25 -6-11=-17$$

The solution is the pair that gives sum -17.

$$a=-11 b=-6$$

Rewrite x^{2}-17x+66 as \left(x^{2}-11x\right)+\left(-6x+66\right).

$$\left(x^{2}-11x\right)+\left(-6x+66\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-11\right)-6\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x-6\right)$$

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.

$$x^{2}-17x+66=0$$

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.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 66}}{2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 66}}{2}$$

Multiply -4 times 66.

$$x=\frac{-\left(-17\right)±\sqrt{289-264}}{2}$$

Add 289 to -264.

$$x=\frac{-\left(-17\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-17\right)±5}{2}$$

The opposite of -17 is 17.

$$x=\frac{17±5}{2}$$

Now solve the equation x=\frac{17±5}{2} when ± is plus. Add 17 to 5.

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

Now solve the equation x=\frac{17±5}{2} when ± is minus. Subtract 5 from 17.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 11 for x_{1} and 6 for x_{2}.

$$x^{2}-17x+66=\left(x-11\right)\left(x-6\right)$$

Evaluate

$$\left(x-11\right)\left(x-6\right)$$