Solve 3a^2-2a-1

$$3 a ^ { 2 } - 2 a - 1$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3a^{2}+pa+qa-1. To find p and q, set up a system to be solved.

$$p+q=-2 pq=3\left(-1\right)=-3$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$p=-3 q=1$$

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

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

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

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

Factor out common term a-1 by using distributive property.

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

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.

$$3a^{2}-2a-1=0$$

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.

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

Square -2.

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

Multiply -4 times 3.

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

Multiply -12 times -1.

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

Add 4 to 12.

$$a=\frac{-\left(-2\right)±\sqrt{16}}{2\times 3}$$

Take the square root of 16.

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

The opposite of -2 is 2.

$$a=\frac{2±4}{2\times 3}$$

Multiply 2 times 3.

$$a=\frac{2±4}{6}$$

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

$$a=\frac{6}{6}$$

Divide 6 by 6.

$$a=1$$

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

$$a=-\frac{2}{6}$$

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

$$a=-\frac{1}{3}$$

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

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

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

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

Add \frac{1}{3} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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

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

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -\frac{2}{3}x -\frac{1}{3} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{2}{3} rs = -\frac{1}{3}$$

Two numbers r and s sum up to \frac{2}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{2}{3} = \frac{1}{3}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{1}{3} - u s = \frac{1}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{3}

$$(\frac{1}{3} - u) (\frac{1}{3} + u) = -\frac{1}{3}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{1}{9} - u^2 = -\frac{1}{3}$$

Simplify the expression by subtracting \frac{1}{9} on both sides

$$-u^2 = -\frac{1}{3}-\frac{1}{9} = -\frac{4}{9}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{4}{9} u = \pm\sqrt{\frac{4}{9}} = \pm \frac{2}{3}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{1}{3} - \frac{2}{3} = -0.333 s = \frac{1}{3} + \frac{2}{3} = 1$$

Evaluate

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