Solve (44y^2-24y+1)

$$( 44 y ^ { 2 } - 24 y + 1 )$$

Factor

$$\left(2y-1\right)\left(22y-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 44y^{2}+ay+by+1. To find a and b, set up a system to be solved.

$$a+b=-24 ab=44\times 1=44$$

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 44.

$$-1,-44 -2,-22 -4,-11$$

Calculate the sum for each pair.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

The solution is the pair that gives sum -24.

$$a=-22 b=-2$$

Rewrite 44y^{2}-24y+1 as \left(44y^{2}-22y\right)+\left(-2y+1\right).

$$\left(44y^{2}-22y\right)+\left(-2y+1\right)$$

Factor out 22y in the first and -1 in the second group.

$$22y\left(2y-1\right)-\left(2y-1\right)$$

Factor out common term 2y-1 by using distributive property.

$$\left(2y-1\right)\left(22y-1\right)$$

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.

$$44y^{2}-24y+1=0$$

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.

$$y=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 44}}{2\times 44}$$

Square -24.

$$y=\frac{-\left(-24\right)±\sqrt{576-4\times 44}}{2\times 44}$$

Multiply -4 times 44.

$$y=\frac{-\left(-24\right)±\sqrt{576-176}}{2\times 44}$$

Add 576 to -176.

$$y=\frac{-\left(-24\right)±\sqrt{400}}{2\times 44}$$

Take the square root of 400.

$$y=\frac{-\left(-24\right)±20}{2\times 44}$$

The opposite of -24 is 24.

$$y=\frac{24±20}{2\times 44}$$

Multiply 2 times 44.

$$y=\frac{24±20}{88}$$

Now solve the equation y=\frac{24±20}{88} when ± is plus. Add 24 to 20.

$$y=\frac{44}{88}$$

Reduce the fraction \frac{44}{88} to lowest terms by extracting and canceling out 44.

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

Now solve the equation y=\frac{24±20}{88} when ± is minus. Subtract 20 from 24.

$$y=\frac{4}{88}$$

Reduce the fraction \frac{4}{88} to lowest terms by extracting and canceling out 4.

$$y=\frac{1}{22}$$

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

$$44y^{2}-24y+1=44\left(y-\frac{1}{2}\right)\left(y-\frac{1}{22}\right)$$

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

$$44y^{2}-24y+1=44\times \frac{2y-1}{2}\left(y-\frac{1}{22}\right)$$

Subtract \frac{1}{22} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$44y^{2}-24y+1=44\times \frac{2y-1}{2}\times \frac{22y-1}{22}$$

Multiply \frac{2y-1}{2} times \frac{22y-1}{22} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$44y^{2}-24y+1=44\times \frac{\left(2y-1\right)\left(22y-1\right)}{2\times 22}$$

Multiply 2 times 22.

$$44y^{2}-24y+1=44\times \frac{\left(2y-1\right)\left(22y-1\right)}{44}$$

Cancel out 44, the greatest common factor in 44 and 44.

$$44y^{2}-24y+1=\left(2y-1\right)\left(22y-1\right)$$

Evaluate

$$\left(2y-1\right)\left(22y-1\right)$$