Solve 2x^2-2x-24

$$2 { x }^{ 2 } -2x-24$$

Factor

$$2\left(x-4\right)\left(x+3\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}-x-12\right)$$

Consider x^{2}-x-12. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-12\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -1.

$$a=-4 b=3$$

Rewrite x^{2}-x-12 as \left(x^{2}-4x\right)+\left(3x-12\right).

$$\left(x^{2}-4x\right)+\left(3x-12\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-4\right)+3\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$2\left(x-4\right)\left(x+3\right)$$

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.

$$2x^{2}-2x-24=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-24\right)}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-24\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-24\right)}}{2\times 2}$$

Multiply -8 times -24.

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

Add 4 to 192.

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

Take the square root of 196.

$$x=\frac{-\left(-2\right)±14}{2\times 2}$$

The opposite of -2 is 2.

$$x=\frac{2±14}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±14}{4}$$

Now solve the equation x=\frac{2±14}{4} when ± is plus. Add 2 to 14.

$$x=\frac{16}{4}$$

Divide 16 by 4.

$$x=4$$

Now solve the equation x=\frac{2±14}{4} when ± is minus. Subtract 14 from 2.

$$x=-\frac{12}{4}$$

Divide -12 by 4.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and -3 for x_{2}.

$$2x^{2}-2x-24=2\left(x-4\right)\left(x-\left(-3\right)\right)$$

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

$$2x^{2}-2x-24=2\left(x-4\right)\left(x+3\right)$$

Evaluate

$$2\left(x-4\right)\left(x+3\right)$$