Solve 3x^2-13x+12

$$3 x ^ { 2 } - 13 x + 12$$

Factor

$$\left(x-3\right)\left(3x-4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=-13 ab=3\times 12=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -13.

$$a=-9 b=-4$$

Rewrite 3x^{2}-13x+12 as \left(3x^{2}-9x\right)+\left(-4x+12\right).

$$\left(3x^{2}-9x\right)+\left(-4x+12\right)$$

Factor out 3x in the first and -4 in the second group.

$$3x\left(x-3\right)-4\left(x-3\right)$$

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

$$\left(x-3\right)\left(3x-4\right)$$

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.

$$3x^{2}-13x+12=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 3\times 12}}{2\times 3}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$x=\frac{-\left(-13\right)±\sqrt{169-144}}{2\times 3}$$

Add 169 to -144.

$$x=\frac{-\left(-13\right)±\sqrt{25}}{2\times 3}$$

Take the square root of 25.

$$x=\frac{-\left(-13\right)±5}{2\times 3}$$

The opposite of -13 is 13.

$$x=\frac{13±5}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{13±5}{6}$$

Now solve the equation x=\frac{13±5}{6} when ± is plus. Add 13 to 5.

$$x=\frac{18}{6}$$

Divide 18 by 6.

$$x=3$$

Now solve the equation x=\frac{13±5}{6} when ± is minus. Subtract 5 from 13.

$$x=\frac{8}{6}$$

Reduce the fraction \frac{8}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and \frac{4}{3} for x_{2}.

$$3x^{2}-13x+12=3\left(x-3\right)\left(x-\frac{4}{3}\right)$$

Subtract \frac{4}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}-13x+12=3\left(x-3\right)\times \frac{3x-4}{3}$$

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

$$3x^{2}-13x+12=\left(x-3\right)\left(3x-4\right)$$

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

$$x ^ 2 -\frac{13}{3}x +4 = 0$$

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

$$r + s = \frac{13}{3} rs = 4$$

Two numbers r and s sum up to \frac{13}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{13}{3} = \frac{13}{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>

$$r = \frac{13}{6} - u s = \frac{13}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 4

$$(\frac{13}{6} - u) (\frac{13}{6} + u) = 4$$

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

$$\frac{169}{36} - u^2 = 4$$

Simplify the expression by subtracting \frac{169}{36} on both sides

$$-u^2 = 4-\frac{169}{36} = -\frac{25}{36}$$

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

$$u^2 = \frac{25}{36} u = \pm\sqrt{\frac{25}{36}} = \pm \frac{5}{6}$$

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

$$r =\frac{13}{6} - \frac{5}{6} = 1.333 s = \frac{13}{6} + \frac{5}{6} = 3.000$$

Evaluate

$$\left(x-3\right)\left(3x-4\right)$$