Solve 9y^2-31y+22

$$9 y ^ { 2 } - 31 y + 22$$

Factor

$$\left(y-1\right)\left(9y-22\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9y^{2}+ay+by+22. To find a and b, set up a system to be solved.

$$a+b=-31 ab=9\times 22=198$$

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 198.

$$-1,-198 -2,-99 -3,-66 -6,-33 -9,-22 -11,-18$$

Calculate the sum for each pair.

$$-1-198=-199 -2-99=-101 -3-66=-69 -6-33=-39 -9-22=-31 -11-18=-29$$

The solution is the pair that gives sum -31.

$$a=-22 b=-9$$

Rewrite 9y^{2}-31y+22 as \left(9y^{2}-22y\right)+\left(-9y+22\right).

$$\left(9y^{2}-22y\right)+\left(-9y+22\right)$$

Factor out y in the first and -1 in the second group.

$$y\left(9y-22\right)-\left(9y-22\right)$$

Factor out common term 9y-22 by using distributive property.

$$\left(9y-22\right)\left(y-1\right)$$

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.

$$9y^{2}-31y+22=0$$

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.

$$y=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 9\times 22}}{2\times 9}$$

Square -31.

$$y=\frac{-\left(-31\right)±\sqrt{961-4\times 9\times 22}}{2\times 9}$$

Multiply -4 times 9.

$$y=\frac{-\left(-31\right)±\sqrt{961-36\times 22}}{2\times 9}$$

Multiply -36 times 22.

$$y=\frac{-\left(-31\right)±\sqrt{961-792}}{2\times 9}$$

Add 961 to -792.

$$y=\frac{-\left(-31\right)±\sqrt{169}}{2\times 9}$$

Take the square root of 169.

$$y=\frac{-\left(-31\right)±13}{2\times 9}$$

The opposite of -31 is 31.

$$y=\frac{31±13}{2\times 9}$$

Multiply 2 times 9.

$$y=\frac{31±13}{18}$$

Now solve the equation y=\frac{31±13}{18} when ± is plus. Add 31 to 13.

$$y=\frac{44}{18}$$

Reduce the fraction \frac{44}{18} to lowest terms by extracting and canceling out 2.

$$y=\frac{22}{9}$$

Now solve the equation y=\frac{31±13}{18} when ± is minus. Subtract 13 from 31.

$$y=\frac{18}{18}$$

Divide 18 by 18.

$$y=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{22}{9} for x_{1} and 1 for x_{2}.

$$9y^{2}-31y+22=9\left(y-\frac{22}{9}\right)\left(y-1\right)$$

Subtract \frac{22}{9} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$9y^{2}-31y+22=9\times \frac{9y-22}{9}\left(y-1\right)$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9y^{2}-31y+22=\left(9y-22\right)\left(y-1\right)$$

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 9

$$x ^ 2 -\frac{31}{9}x +\frac{22}{9} = 0$$

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

$$r + s = \frac{31}{9} rs = \frac{22}{9}$$

Two numbers r and s sum up to \frac{31}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{31}{9} = \frac{31}{18}. 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>

$$r = \frac{31}{18} - u s = \frac{31}{18} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{22}{9}

$$(\frac{31}{18} - u) (\frac{31}{18} + u) = \frac{22}{9}$$

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

$$\frac{961}{324} - u^2 = \frac{22}{9}$$

Simplify the expression by subtracting \frac{961}{324} on both sides

$$-u^2 = \frac{22}{9}-\frac{961}{324} = -\frac{169}{324}$$

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

$$u^2 = \frac{169}{324} u = \pm\sqrt{\frac{169}{324}} = \pm \frac{13}{18}$$

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

$$r =\frac{31}{18} - \frac{13}{18} = 1.000 s = \frac{31}{18} + \frac{13}{18} = 2.444$$

Evaluate

$$\left(y-1\right)\left(9y-22\right)$$