Solve 2x+3x^2-1

$$2 x + 3 x ^ { 2 } - 1$$

Factor

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

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$3x^{2}+2x-1$$

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=2 ab=3\left(-1\right)=-3$$

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. The only such pair is the system solution.

$$a=-1 b=3$$

Rewrite 3x^{2}+2x-1 as \left(3x^{2}-x\right)+\left(3x-1\right).

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

Factor out x in 3x^{2}-x.

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

Factor out common term 3x-1 by using distributive property.

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

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.

$$3x^{2}+2x-1=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 3\left(-1\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-2±\sqrt{4-12\left(-1\right)}}{2\times 3}$$

Multiply -12 times -1.

$$x=\frac{-2±\sqrt{4+12}}{2\times 3}$$

Add 4 to 12.

$$x=\frac{-2±\sqrt{16}}{2\times 3}$$

Take the square root of 16.

$$x=\frac{-2±4}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-2±4}{6}$$

Now solve the equation x=\frac{-2±4}{6} when ± is plus. Add -2 to 4.

$$x=\frac{2}{6}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-2±4}{6} when ± is minus. Subtract 4 from -2.

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

Divide -6 by 6.

$$x=-1$$

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

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

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

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

Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Cancel out 3, the greatest common factor in 3 and 3.

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

Evaluate

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