Solve x^2+32x-273

$${ x }^{ 2 } +32x-273$$

Factor

$$\left(x-7\right)\left(x+39\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-273. To find a and b, set up a system to be solved.

$$a+b=32 ab=1\left(-273\right)=-273$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -273.

$$-1,273 -3,91 -7,39 -13,21$$

Calculate the sum for each pair.

$$-1+273=272 -3+91=88 -7+39=32 -13+21=8$$

The solution is the pair that gives sum 32.

$$a=-7 b=39$$

Rewrite x^{2}+32x-273 as \left(x^{2}-7x\right)+\left(39x-273\right).

$$\left(x^{2}-7x\right)+\left(39x-273\right)$$

Factor out x in the first and 39 in the second group.

$$x\left(x-7\right)+39\left(x-7\right)$$

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

$$\left(x-7\right)\left(x+39\right)$$

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.

$$x^{2}+32x-273=0$$

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.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-273\right)}}{2}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-273\right)}}{2}$$

Multiply -4 times -273.

$$x=\frac{-32±\sqrt{1024+1092}}{2}$$

Add 1024 to 1092.

$$x=\frac{-32±\sqrt{2116}}{2}$$

Take the square root of 2116.

$$x=\frac{-32±46}{2}$$

Now solve the equation x=\frac{-32±46}{2} when ± is plus. Add -32 to 46.

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

Divide 14 by 2.

$$x=7$$

Now solve the equation x=\frac{-32±46}{2} when ± is minus. Subtract 46 from -32.

$$x=-\frac{78}{2}$$

Divide -78 by 2.

$$x=-39$$

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 -39 for x_{2}.

$$x^{2}+32x-273=\left(x-7\right)\left(x-\left(-39\right)\right)$$

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

$$x^{2}+32x-273=\left(x-7\right)\left(x+39\right)$$

Evaluate

$$\left(x-7\right)\left(x+39\right)$$