Solve 12x^2-13x+1

$$12 x ^ { 2 } - 13 x + 1$$

Factor

$$\left(x-1\right)\left(12x-1\right)$$

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.

$$a+b=-13 ab=12\times 1=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -13.

$$a=-12 b=-1$$

Rewrite 12x^{2}-13x+1 as \left(12x^{2}-12x\right)+\left(-x+1\right).

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

Factor out 12x in the first and -1 in the second group.

$$12x\left(x-1\right)-\left(x-1\right)$$

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

$$\left(x-1\right)\left(12x-1\right)$$

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.

$$12x^{2}-13x+1=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 12}}{2\times 12}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 12}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-13\right)±\sqrt{169-48}}{2\times 12}$$

Add 169 to -48.

$$x=\frac{-\left(-13\right)±\sqrt{121}}{2\times 12}$$

Take the square root of 121.

$$x=\frac{-\left(-13\right)±11}{2\times 12}$$

The opposite of -13 is 13.

$$x=\frac{13±11}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{13±11}{24}$$

Now solve the equation x=\frac{13±11}{24} when ± is plus. Add 13 to 11.

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

Divide 24 by 24.

$$x=1$$

Now solve the equation x=\frac{13±11}{24} when ± is minus. Subtract 11 from 13.

$$x=\frac{2}{24}$$

Reduce the fraction \frac{2}{24} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{12}$$

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}{12} for x_{2}.

$$12x^{2}-13x+1=12\left(x-1\right)\left(x-\frac{1}{12}\right)$$

Subtract \frac{1}{12} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$12x^{2}-13x+1=12\left(x-1\right)\times \frac{12x-1}{12}$$

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

$$12x^{2}-13x+1=\left(x-1\right)\left(12x-1\right)$$

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

$$x ^ 2 -\frac{13}{12}x +\frac{1}{12} = 0$$

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

$$r + s = \frac{13}{12} rs = \frac{1}{12}$$

Two numbers r and s sum up to \frac{13}{12} exactly when the average of the two numbers is \frac{1}{2}*\frac{13}{12} = \frac{13}{24}. 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>

$$r = \frac{13}{24} - u s = \frac{13}{24} + u$$

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

$$(\frac{13}{24} - u) (\frac{13}{24} + u) = \frac{1}{12}$$

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

$$\frac{169}{576} - u^2 = \frac{1}{12}$$

Simplify the expression by subtracting \frac{169}{576} on both sides

$$-u^2 = \frac{1}{12}-\frac{169}{576} = -\frac{121}{576}$$

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

$$u^2 = \frac{121}{576} u = \pm\sqrt{\frac{121}{576}} = \pm \frac{11}{24}$$

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

$$r =\frac{13}{24} - \frac{11}{24} = 0.083 s = \frac{13}{24} + \frac{11}{24} = 1$$

Evaluate

$$\left(x-1\right)\left(12x-1\right)$$