Solve 12x^2-4x-1

$$12 x ^ { 2 } - 4 x - 1$$

Factor

$$\left(2x-1\right)\left(6x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 12x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-4 ab=12\left(-1\right)=-12$$

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 -4.

$$a=-6 b=2$$

Rewrite 12x^{2}-4x-1 as \left(12x^{2}-6x\right)+\left(2x-1\right).

$$\left(12x^{2}-6x\right)+\left(2x-1\right)$$

Factor out 6x in 12x^{2}-6x.

$$6x\left(2x-1\right)+2x-1$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(6x+1\right)$$

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.

$$12x^{2}-4x-1=0$$

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.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 12\left(-1\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-4\right)±\sqrt{16-48\left(-1\right)}}{2\times 12}$$

Multiply -48 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+48}}{2\times 12}$$

Add 16 to 48.

$$x=\frac{-\left(-4\right)±\sqrt{64}}{2\times 12}$$

Take the square root of 64.

$$x=\frac{-\left(-4\right)±8}{2\times 12}$$

The opposite of -4 is 4.

$$x=\frac{4±8}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{4±8}{24}$$

Now solve the equation x=\frac{4±8}{24} when ± is plus. Add 4 to 8.

$$x=\frac{12}{24}$$

Reduce the fraction \frac{12}{24} to lowest terms by extracting and canceling out 12.

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

Now solve the equation x=\frac{4±8}{24} when ± is minus. Subtract 8 from 4.

$$x=-\frac{4}{24}$$

Reduce the fraction \frac{-4}{24} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2} for x_{1} and -\frac{1}{6} for x_{2}.

$$12x^{2}-4x-1=12\left(x-\frac{1}{2}\right)\left(x-\left(-\frac{1}{6}\right)\right)$$

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

$$12x^{2}-4x-1=12\left(x-\frac{1}{2}\right)\left(x+\frac{1}{6}\right)$$

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

$$12x^{2}-4x-1=12\times \frac{2x-1}{2}\left(x+\frac{1}{6}\right)$$

Add \frac{1}{6} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$12x^{2}-4x-1=12\times \frac{2x-1}{2}\times \frac{6x+1}{6}$$

Multiply \frac{2x-1}{2} times \frac{6x+1}{6} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$12x^{2}-4x-1=12\times \frac{\left(2x-1\right)\left(6x+1\right)}{2\times 6}$$

Multiply 2 times 6.

$$12x^{2}-4x-1=12\times \frac{\left(2x-1\right)\left(6x+1\right)}{12}$$

Cancel out 12, the greatest common factor in 12 and 12.

$$12x^{2}-4x-1=\left(2x-1\right)\left(6x+1\right)$$

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 12

$$x ^ 2 -\frac{1}{3}x -\frac{1}{12} = 0$$

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

$$r + s = \frac{1}{3} rs = -\frac{1}{12}$$

Two numbers r and s sum up to \frac{1}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{1}{3} = \frac{1}{6}. 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>

$$r = \frac{1}{6} - u s = \frac{1}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{12}

$$(\frac{1}{6} - u) (\frac{1}{6} + u) = -\frac{1}{12}$$

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

$$\frac{1}{36} - u^2 = -\frac{1}{12}$$

Simplify the expression by subtracting \frac{1}{36} on both sides

$$-u^2 = -\frac{1}{12}-\frac{1}{36} = -\frac{1}{9}$$

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

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

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

$$r =\frac{1}{6} - \frac{1}{3} = -0.167 s = \frac{1}{6} + \frac{1}{3} = 0.500$$

Evaluate

$$\left(2x-1\right)\left(6x+1\right)$$