Solve 8x^2-52x+24

$$8 x ^ { 2 } - 52 x + 24$$

Factor

$$4\left(x-6\right)\left(2x-1\right)$$

Solution Steps

Factor out 4.

$$4\left(2x^{2}-13x+6\right)$$

Consider 2x^{2}-13x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-13 ab=2\times 6=12$$

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 2x^{2}-13x+6 as \left(2x^{2}-12x\right)+\left(-x+6\right).

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

Factor out 2x in the first and -1 in the second group.

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

Factor out common term x-6 by using distributive property.

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

Rewrite the complete factored expression.

$$4\left(x-6\right)\left(2x-1\right)$$

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.

$$8x^{2}-52x+24=0$$

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.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 8\times 24}}{2\times 8}$$

Square -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 8\times 24}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-52\right)±\sqrt{2704-32\times 24}}{2\times 8}$$

Multiply -32 times 24.

$$x=\frac{-\left(-52\right)±\sqrt{2704-768}}{2\times 8}$$

Add 2704 to -768.

$$x=\frac{-\left(-52\right)±\sqrt{1936}}{2\times 8}$$

Take the square root of 1936.

$$x=\frac{-\left(-52\right)±44}{2\times 8}$$

The opposite of -52 is 52.

$$x=\frac{52±44}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{52±44}{16}$$

Now solve the equation x=\frac{52±44}{16} when ± is plus. Add 52 to 44.

$$x=\frac{96}{16}$$

Divide 96 by 16.

$$x=6$$

Now solve the equation x=\frac{52±44}{16} when ± is minus. Subtract 44 from 52.

$$x=\frac{8}{16}$$

Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.

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

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

$$8x^{2}-52x+24=8\left(x-6\right)\left(x-\frac{1}{2}\right)$$

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

$$8x^{2}-52x+24=8\left(x-6\right)\times \frac{2x-1}{2}$$

Cancel out 2, the greatest common factor in 8 and 2.

$$8x^{2}-52x+24=4\left(x-6\right)\left(2x-1\right)$$

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 8

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

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

$$r + s = \frac{13}{2} rs = 3$$

Two numbers r and s sum up to \frac{13}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{13}{2} = \frac{13}{4}. 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>

$$r = \frac{13}{4} - u s = \frac{13}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 3

$$(\frac{13}{4} - u) (\frac{13}{4} + u) = 3$$

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

$$\frac{169}{16} - u^2 = 3$$

Simplify the expression by subtracting \frac{169}{16} on both sides

$$-u^2 = 3-\frac{169}{16} = -\frac{121}{16}$$

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

$$u^2 = \frac{121}{16} u = \pm\sqrt{\frac{121}{16}} = \pm \frac{11}{4}$$

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

$$r =\frac{13}{4} - \frac{11}{4} = 0.500 s = \frac{13}{4} + \frac{11}{4} = 6$$

Evaluate

$$4\left(x-6\right)\left(2x-1\right)$$