Solve x^2-11x-26

$${ x }^{ 2 } -11x-26$$

Factor

$$\left(x-13\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-26. To find a and b, set up a system to be solved.

$$a+b=-11 ab=1\left(-26\right)=-26$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -26.

$$1,-26 2,-13$$

Calculate the sum for each pair.

$$1-26=-25 2-13=-11$$

The solution is the pair that gives sum -11.

$$a=-13 b=2$$

Rewrite x^{2}-11x-26 as \left(x^{2}-13x\right)+\left(2x-26\right).

$$\left(x^{2}-13x\right)+\left(2x-26\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(x-13\right)+2\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x+2\right)$$

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.

$$x^{2}-11x-26=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-26\right)}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-26\right)}}{2}$$

Multiply -4 times -26.

$$x=\frac{-\left(-11\right)±\sqrt{121+104}}{2}$$

Add 121 to 104.

$$x=\frac{-\left(-11\right)±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-\left(-11\right)±15}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±15}{2}$$

Now solve the equation x=\frac{11±15}{2} when ± is plus. Add 11 to 15.

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

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{11±15}{2} when ± is minus. Subtract 15 from 11.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13 for x_{1} and -2 for x_{2}.

$$x^{2}-11x-26=\left(x-13\right)\left(x-\left(-2\right)\right)$$

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

$$x^{2}-11x-26=\left(x-13\right)\left(x+2\right)$$

Evaluate

$$\left(x-13\right)\left(x+2\right)$$