Solve 8x^2-22x+12

$$8 x ^ { 2 } - 22 x + 12$$

Factor

$$2\left(x-2\right)\left(4x-3\right)$$

Solution Steps

Factor out 2.

$$2\left(4x^{2}-11x+6\right)$$

Consider 4x^{2}-11x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-11 ab=4\times 6=24$$

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 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-8 b=-3$$

Rewrite 4x^{2}-11x+6 as \left(4x^{2}-8x\right)+\left(-3x+6\right).

$$\left(4x^{2}-8x\right)+\left(-3x+6\right)$$

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

$$4x\left(x-2\right)-3\left(x-2\right)$$

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

$$\left(x-2\right)\left(4x-3\right)$$

Rewrite the complete factored expression.

$$2\left(x-2\right)\left(4x-3\right)$$

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.

$$8x^{2}-22x+12=0$$

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.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\times 12}}{2\times 8}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 8\times 12}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-22\right)±\sqrt{484-32\times 12}}{2\times 8}$$

Multiply -32 times 12.

$$x=\frac{-\left(-22\right)±\sqrt{484-384}}{2\times 8}$$

Add 484 to -384.

$$x=\frac{-\left(-22\right)±\sqrt{100}}{2\times 8}$$

Take the square root of 100.

$$x=\frac{-\left(-22\right)±10}{2\times 8}$$

The opposite of -22 is 22.

$$x=\frac{22±10}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{22±10}{16}$$

Now solve the equation x=\frac{22±10}{16} when ± is plus. Add 22 to 10.

$$x=\frac{32}{16}$$

Divide 32 by 16.

$$x=2$$

Now solve the equation x=\frac{22±10}{16} when ± is minus. Subtract 10 from 22.

$$x=\frac{12}{16}$$

Reduce the fraction \frac{12}{16} to lowest terms by extracting and canceling out 4.

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

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

$$8x^{2}-22x+12=8\left(x-2\right)\left(x-\frac{3}{4}\right)$$

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

$$8x^{2}-22x+12=8\left(x-2\right)\times \frac{4x-3}{4}$$

Cancel out 4, the greatest common factor in 8 and 4.

$$8x^{2}-22x+12=2\left(x-2\right)\left(4x-3\right)$$

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

$$x ^ 2 -\frac{11}{4}x +\frac{3}{2} = 0$$

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

$$r + s = \frac{11}{4} rs = \frac{3}{2}$$

Two numbers r and s sum up to \frac{11}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{11}{4} = \frac{11}{8}. 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>

$$r = \frac{11}{8} - u s = \frac{11}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{2}

$$(\frac{11}{8} - u) (\frac{11}{8} + u) = \frac{3}{2}$$

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

$$\frac{121}{64} - u^2 = \frac{3}{2}$$

Simplify the expression by subtracting \frac{121}{64} on both sides

$$-u^2 = \frac{3}{2}-\frac{121}{64} = -\frac{25}{64}$$

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

$$u^2 = \frac{25}{64} u = \pm\sqrt{\frac{25}{64}} = \pm \frac{5}{8}$$

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

$$r =\frac{11}{8} - \frac{5}{8} = 0.750 s = \frac{11}{8} + \frac{5}{8} = 2$$

Evaluate

$$2\left(x-2\right)\left(4x-3\right)$$