Solve x^2-44x+363

$${ x }^{ 2 } -44x+363$$

Factor

$$\left(x-33\right)\left(x-11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+363. To find a and b, set up a system to be solved.

$$a+b=-44 ab=1\times 363=363$$

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 363.

$$-1,-363 -3,-121 -11,-33$$

Calculate the sum for each pair.

$$-1-363=-364 -3-121=-124 -11-33=-44$$

The solution is the pair that gives sum -44.

$$a=-33 b=-11$$

Rewrite x^{2}-44x+363 as \left(x^{2}-33x\right)+\left(-11x+363\right).

$$\left(x^{2}-33x\right)+\left(-11x+363\right)$$

Factor out x in the first and -11 in the second group.

$$x\left(x-33\right)-11\left(x-33\right)$$

Factor out common term x-33 by using distributive property.

$$\left(x-33\right)\left(x-11\right)$$

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.

$$x^{2}-44x+363=0$$

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.

$$x=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 363}}{2}$$

Square -44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-4\times 363}}{2}$$

Multiply -4 times 363.

$$x=\frac{-\left(-44\right)±\sqrt{1936-1452}}{2}$$

Add 1936 to -1452.

$$x=\frac{-\left(-44\right)±\sqrt{484}}{2}$$

Take the square root of 484.

$$x=\frac{-\left(-44\right)±22}{2}$$

The opposite of -44 is 44.

$$x=\frac{44±22}{2}$$

Now solve the equation x=\frac{44±22}{2} when ± is plus. Add 44 to 22.

$$x=\frac{66}{2}$$

Divide 66 by 2.

$$x=33$$

Now solve the equation x=\frac{44±22}{2} when ± is minus. Subtract 22 from 44.

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

Divide 22 by 2.

$$x=11$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 33 for x_{1} and 11 for x_{2}.

$$x^{2}-44x+363=\left(x-33\right)\left(x-11\right)$$

Evaluate

$$\left(x-33\right)\left(x-11\right)$$