Solve 8d^2-22d+9

$$8 d ^ { 2 } - 22 d + 9$$

Factor

$$\left(2d-1\right)\left(4d-9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8d^{2}+ad+bd+9. To find a and b, set up a system to be solved.

$$a+b=-22 ab=8\times 9=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -22.

$$a=-18 b=-4$$

Rewrite 8d^{2}-22d+9 as \left(8d^{2}-18d\right)+\left(-4d+9\right).

$$\left(8d^{2}-18d\right)+\left(-4d+9\right)$$

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

$$2d\left(4d-9\right)-\left(4d-9\right)$$

Factor out common term 4d-9 by using distributive property.

$$\left(4d-9\right)\left(2d-1\right)$$

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.

$$8d^{2}-22d+9=0$$

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.

$$d=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\times 9}}{2\times 8}$$

Square -22.

$$d=\frac{-\left(-22\right)±\sqrt{484-4\times 8\times 9}}{2\times 8}$$

Multiply -4 times 8.

$$d=\frac{-\left(-22\right)±\sqrt{484-32\times 9}}{2\times 8}$$

Multiply -32 times 9.

$$d=\frac{-\left(-22\right)±\sqrt{484-288}}{2\times 8}$$

Add 484 to -288.

$$d=\frac{-\left(-22\right)±\sqrt{196}}{2\times 8}$$

Take the square root of 196.

$$d=\frac{-\left(-22\right)±14}{2\times 8}$$

The opposite of -22 is 22.

$$d=\frac{22±14}{2\times 8}$$

Multiply 2 times 8.

$$d=\frac{22±14}{16}$$

Now solve the equation d=\frac{22±14}{16} when ± is plus. Add 22 to 14.

$$d=\frac{36}{16}$$

Reduce the fraction \frac{36}{16} to lowest terms by extracting and canceling out 4.

$$d=\frac{9}{4}$$

Now solve the equation d=\frac{22±14}{16} when ± is minus. Subtract 14 from 22.

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

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

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{4} for x_{1} and \frac{1}{2} for x_{2}.

$$8d^{2}-22d+9=8\left(d-\frac{9}{4}\right)\left(d-\frac{1}{2}\right)$$

Subtract \frac{9}{4} from d by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8d^{2}-22d+9=8\times \frac{4d-9}{4}\left(d-\frac{1}{2}\right)$$

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

$$8d^{2}-22d+9=8\times \frac{4d-9}{4}\times \frac{2d-1}{2}$$

Multiply \frac{4d-9}{4} times \frac{2d-1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8d^{2}-22d+9=8\times \frac{\left(4d-9\right)\left(2d-1\right)}{4\times 2}$$

Multiply 4 times 2.

$$8d^{2}-22d+9=8\times \frac{\left(4d-9\right)\left(2d-1\right)}{8}$$

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

$$8d^{2}-22d+9=\left(4d-9\right)\left(2d-1\right)$$

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

$$x ^ 2 -\frac{11}{4}x +\frac{9}{8} = 0$$

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

$$r + s = \frac{11}{4} rs = \frac{9}{8}$$

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{9}{8}

$$(\frac{11}{8} - u) (\frac{11}{8} + u) = \frac{9}{8}$$

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

$$\frac{121}{64} - u^2 = \frac{9}{8}$$

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

$$-u^2 = \frac{9}{8}-\frac{121}{64} = -\frac{49}{64}$$

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

$$u^2 = \frac{49}{64} u = \pm\sqrt{\frac{49}{64}} = \pm \frac{7}{8}$$

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

$$r =\frac{11}{8} - \frac{7}{8} = 0.500 s = \frac{11}{8} + \frac{7}{8} = 2.250$$

Evaluate

$$\left(2d-1\right)\left(4d-9\right)$$