Solve x^2+-18x+77

$${ x }^{ 2 } +-18x+77$$

Factor

$$\left(x-11\right)\left(x-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+77. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 77=77$$

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 77.

$$-1,-77 -7,-11$$

Calculate the sum for each pair.

$$-1-77=-78 -7-11=-18$$

The solution is the pair that gives sum -18.

$$a=-11 b=-7$$

Rewrite x^{2}-18x+77 as \left(x^{2}-11x\right)+\left(-7x+77\right).

$$\left(x^{2}-11x\right)+\left(-7x+77\right)$$

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

$$x\left(x-11\right)-7\left(x-11\right)$$

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

$$\left(x-11\right)\left(x-7\right)$$

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.

$$x^{2}-18x+77=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 77}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 77}}{2}$$

Multiply -4 times 77.

$$x=\frac{-\left(-18\right)±\sqrt{324-308}}{2}$$

Add 324 to -308.

$$x=\frac{-\left(-18\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-18\right)±4}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±4}{2}$$

Now solve the equation x=\frac{18±4}{2} when ± is plus. Add 18 to 4.

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

Divide 22 by 2.

$$x=11$$

Now solve the equation x=\frac{18±4}{2} when ± is minus. Subtract 4 from 18.

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

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

$$x^{2}-18x+77=\left(x-11\right)\left(x-7\right)$$

Evaluate

$$\left(x-11\right)\left(x-7\right)$$