Solve 6x^2-x-2

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

Factor

$$\left(3x-2\right)\left(2x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-1 ab=6\left(-2\right)=-12$$

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 6x^{2}-x-2 as \left(6x^{2}-4x\right)+\left(3x-2\right).

$$\left(6x^{2}-4x\right)+\left(3x-2\right)$$

Factor out 2x in 6x^{2}-4x.

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

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(2x+1\right)$$

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.

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

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 6\left(-2\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-1\right)±\sqrt{1-24\left(-2\right)}}{2\times 6}$$

Multiply -24 times -2.

$$x=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times 6}$$

Add 1 to 48.

$$x=\frac{-\left(-1\right)±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$x=\frac{-\left(-1\right)±7}{2\times 6}$$

The opposite of -1 is 1.

$$x=\frac{1±7}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{1±7}{12}$$

Now solve the equation x=\frac{1±7}{12} when ± is plus. Add 1 to 7.

$$x=\frac{8}{12}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{1±7}{12} when ± is minus. Subtract 7 from 1.

$$x=-\frac{6}{12}$$

Reduce the fraction \frac{-6}{12} to lowest terms by extracting and canceling out 6.

$$x=-\frac{1}{2}$$

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

$$6x^{2}-x-2=6\left(x-\frac{2}{3}\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$6x^{2}-x-2=6\left(x-\frac{2}{3}\right)\left(x+\frac{1}{2}\right)$$

Subtract \frac{2}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}-x-2=6\times \frac{3x-2}{3}\left(x+\frac{1}{2}\right)$$

Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}-x-2=6\times \frac{3x-2}{3}\times \frac{2x+1}{2}$$

Multiply \frac{3x-2}{3} times \frac{2x+1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$6x^{2}-x-2=6\times \frac{\left(3x-2\right)\left(2x+1\right)}{3\times 2}$$

Multiply 3 times 2.

$$6x^{2}-x-2=6\times \frac{\left(3x-2\right)\left(2x+1\right)}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6x^{2}-x-2=\left(3x-2\right)\left(2x+1\right)$$

Evaluate

$$\left(3x-2\right)\left(2x+1\right)$$