Solve x^2-8x-65

$${ x }^{ 2 } -8x-65$$

Factor

$$\left(x-13\right)\left(x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-65. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-65\right)=-65$$

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 -65.

$$1,-65 5,-13$$

Calculate the sum for each pair.

$$1-65=-64 5-13=-8$$

The solution is the pair that gives sum -8.

$$a=-13 b=5$$

Rewrite x^{2}-8x-65 as \left(x^{2}-13x\right)+\left(5x-65\right).

$$\left(x^{2}-13x\right)+\left(5x-65\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x-13\right)+5\left(x-13\right)$$

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

$$\left(x-13\right)\left(x+5\right)$$

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.

$$x^{2}-8x-65=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-65\right)}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-65\right)}}{2}$$

Multiply -4 times -65.

$$x=\frac{-\left(-8\right)±\sqrt{64+260}}{2}$$

Add 64 to 260.

$$x=\frac{-\left(-8\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-8\right)±18}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±18}{2}$$

Now solve the equation x=\frac{8±18}{2} when ± is plus. Add 8 to 18.

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

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{8±18}{2} when ± is minus. Subtract 18 from 8.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

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 -5 for x_{2}.

$$x^{2}-8x-65=\left(x-13\right)\left(x-\left(-5\right)\right)$$

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

$$x^{2}-8x-65=\left(x-13\right)\left(x+5\right)$$

Evaluate

$$\left(x-13\right)\left(x+5\right)$$