Solve 14x^2+3x-2

$$14 { x }^{ 2 } +3x-2$$

Factor

$$\left(7x-2\right)\left(2x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 14x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=3 ab=14\left(-2\right)=-28$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -28.

$$-1,28 -2,14 -4,7$$

Calculate the sum for each pair.

$$-1+28=27 -2+14=12 -4+7=3$$

The solution is the pair that gives sum 3.

$$a=-4 b=7$$

Rewrite 14x^{2}+3x-2 as \left(14x^{2}-4x\right)+\left(7x-2\right).

$$\left(14x^{2}-4x\right)+\left(7x-2\right)$$

Factor out 2x in 14x^{2}-4x.

$$2x\left(7x-2\right)+7x-2$$

Factor out common term 7x-2 by using distributive property.

$$\left(7x-2\right)\left(2x+1\right)$$

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.

$$14x^{2}+3x-2=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 14\left(-2\right)}}{2\times 14}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 14\left(-2\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-3±\sqrt{9-56\left(-2\right)}}{2\times 14}$$

Multiply -56 times -2.

$$x=\frac{-3±\sqrt{9+112}}{2\times 14}$$

Add 9 to 112.

$$x=\frac{-3±\sqrt{121}}{2\times 14}$$

Take the square root of 121.

$$x=\frac{-3±11}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{-3±11}{28}$$

Now solve the equation x=\frac{-3±11}{28} when ± is plus. Add -3 to 11.

$$x=\frac{8}{28}$$

Reduce the fraction \frac{8}{28} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{7}$$

Now solve the equation x=\frac{-3±11}{28} when ± is minus. Subtract 11 from -3.

$$x=-\frac{14}{28}$$

Reduce the fraction \frac{-14}{28} to lowest terms by extracting and canceling out 14.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{7} for x_{1} and -\frac{1}{2} for x_{2}.

$$14x^{2}+3x-2=14\left(x-\frac{2}{7}\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$14x^{2}+3x-2=14\left(x-\frac{2}{7}\right)\left(x+\frac{1}{2}\right)$$

Subtract \frac{2}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$14x^{2}+3x-2=14\times \frac{7x-2}{7}\left(x+\frac{1}{2}\right)$$

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

$$14x^{2}+3x-2=14\times \frac{7x-2}{7}\times \frac{2x+1}{2}$$

Multiply \frac{7x-2}{7} times \frac{2x+1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$14x^{2}+3x-2=14\times \frac{\left(7x-2\right)\left(2x+1\right)}{7\times 2}$$

Multiply 7 times 2.

$$14x^{2}+3x-2=14\times \frac{\left(7x-2\right)\left(2x+1\right)}{14}$$

Cancel out 14, the greatest common factor in 14 and 14.

$$14x^{2}+3x-2=\left(7x-2\right)\left(2x+1\right)$$

Evaluate

$$\left(7x-2\right)\left(2x+1\right)$$