Solve x^2+6x-91

$${ x }^{ 2 } +6x-91$$

Factor

$$\left(x-7\right)\left(x+13\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-91. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-91\right)=-91$$

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 -91.

$$-1,91 -7,13$$

Calculate the sum for each pair.

$$-1+91=90 -7+13=6$$

The solution is the pair that gives sum 6.

$$a=-7 b=13$$

Rewrite x^{2}+6x-91 as \left(x^{2}-7x\right)+\left(13x-91\right).

$$\left(x^{2}-7x\right)+\left(13x-91\right)$$

Factor out x in the first and 13 in the second group.

$$x\left(x-7\right)+13\left(x-7\right)$$

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

$$\left(x-7\right)\left(x+13\right)$$

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.

$$x^{2}+6x-91=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-91\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-91\right)}}{2}$$

Multiply -4 times -91.

$$x=\frac{-6±\sqrt{36+364}}{2}$$

Add 36 to 364.

$$x=\frac{-6±\sqrt{400}}{2}$$

Take the square root of 400.

$$x=\frac{-6±20}{2}$$

Now solve the equation x=\frac{-6±20}{2} when ± is plus. Add -6 to 20.

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

Divide 14 by 2.

$$x=7$$

Now solve the equation x=\frac{-6±20}{2} when ± is minus. Subtract 20 from -6.

$$x=-\frac{26}{2}$$

Divide -26 by 2.

$$x=-13$$

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 -13 for x_{2}.

$$x^{2}+6x-91=\left(x-7\right)\left(x-\left(-13\right)\right)$$

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

$$x^{2}+6x-91=\left(x-7\right)\left(x+13\right)$$

Evaluate

$$\left(x-7\right)\left(x+13\right)$$