Solve 8u^2-22u+5

$$8 u ^ { 2 } - 22 u + 5$$

Factor

$$\left(2u-5\right)\left(4u-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8u^{2}+au+bu+5. To find a and b, set up a system to be solved.

$$a+b=-22 ab=8\times 5=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Calculate the sum for each pair.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

The solution is the pair that gives sum -22.

$$a=-20 b=-2$$

Rewrite 8u^{2}-22u+5 as \left(8u^{2}-20u\right)+\left(-2u+5\right).

$$\left(8u^{2}-20u\right)+\left(-2u+5\right)$$

Factor out 4u in the first and -1 in the second group.

$$4u\left(2u-5\right)-\left(2u-5\right)$$

Factor out common term 2u-5 by using distributive property.

$$\left(2u-5\right)\left(4u-1\right)$$

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.

$$8u^{2}-22u+5=0$$

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.

$$u=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\times 5}}{2\times 8}$$

Square -22.

$$u=\frac{-\left(-22\right)±\sqrt{484-4\times 8\times 5}}{2\times 8}$$

Multiply -4 times 8.

$$u=\frac{-\left(-22\right)±\sqrt{484-32\times 5}}{2\times 8}$$

Multiply -32 times 5.

$$u=\frac{-\left(-22\right)±\sqrt{484-160}}{2\times 8}$$

Add 484 to -160.

$$u=\frac{-\left(-22\right)±\sqrt{324}}{2\times 8}$$

Take the square root of 324.

$$u=\frac{-\left(-22\right)±18}{2\times 8}$$

The opposite of -22 is 22.

$$u=\frac{22±18}{2\times 8}$$

Multiply 2 times 8.

$$u=\frac{22±18}{16}$$

Now solve the equation u=\frac{22±18}{16} when ± is plus. Add 22 to 18.

$$u=\frac{40}{16}$$

Reduce the fraction \frac{40}{16} to lowest terms by extracting and canceling out 8.

$$u=\frac{5}{2}$$

Now solve the equation u=\frac{22±18}{16} when ± is minus. Subtract 18 from 22.

$$u=\frac{4}{16}$$

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

$$u=\frac{1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{2} for x_{1} and \frac{1}{4} for x_{2}.

$$8u^{2}-22u+5=8\left(u-\frac{5}{2}\right)\left(u-\frac{1}{4}\right)$$

Subtract \frac{5}{2} from u by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8u^{2}-22u+5=8\times \frac{2u-5}{2}\left(u-\frac{1}{4}\right)$$

Subtract \frac{1}{4} from u by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8u^{2}-22u+5=8\times \frac{2u-5}{2}\times \frac{4u-1}{4}$$

Multiply \frac{2u-5}{2} times \frac{4u-1}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8u^{2}-22u+5=8\times \frac{\left(2u-5\right)\left(4u-1\right)}{2\times 4}$$

Multiply 2 times 4.

$$8u^{2}-22u+5=8\times \frac{\left(2u-5\right)\left(4u-1\right)}{8}$$

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

$$8u^{2}-22u+5=\left(2u-5\right)\left(4u-1\right)$$

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

$$x ^ 2 -\frac{11}{4}x +\frac{5}{8} = 0$$

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

$$r + s = \frac{11}{4} rs = \frac{5}{8}$$

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{5}{8}

$$(\frac{11}{8} - u) (\frac{11}{8} + u) = \frac{5}{8}$$

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

$$\frac{121}{64} - u^2 = \frac{5}{8}$$

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

$$-u^2 = \frac{5}{8}-\frac{121}{64} = -\frac{81}{64}$$

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

$$u^2 = \frac{81}{64} u = \pm\sqrt{\frac{81}{64}} = \pm \frac{9}{8}$$

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

$$r =\frac{11}{8} - \frac{9}{8} = 0.250 s = \frac{11}{8} + \frac{9}{8} = 2.500$$

Evaluate

$$\left(2u-5\right)\left(4u-1\right)$$