Solve x^2-2x-48

$${ x }^{ 2 } -2x-48$$

Factor

$$\left(x-8\right)\left(x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-48\right)=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -2.

$$a=-8 b=6$$

Rewrite x^{2}-2x-48 as \left(x^{2}-8x\right)+\left(6x-48\right).

$$\left(x^{2}-8x\right)+\left(6x-48\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x-8\right)+6\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+6\right)$$

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.

$$x^{2}-2x-48=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-48\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$x=\frac{-\left(-2\right)±\sqrt{4+192}}{2}$$

Add 4 to 192.

$$x=\frac{-\left(-2\right)±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-\left(-2\right)±14}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±14}{2}$$

Now solve the equation x=\frac{2±14}{2} when ± is plus. Add 2 to 14.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{2±14}{2} when ± is minus. Subtract 14 from 2.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -6 for x_{2}.

$$x^{2}-2x-48=\left(x-8\right)\left(x-\left(-6\right)\right)$$

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

$$x^{2}-2x-48=\left(x-8\right)\left(x+6\right)$$

Evaluate

$$\left(x-8\right)\left(x+6\right)$$