Solve 8y^2+22y+5

$$8 { y }^{ 2 } +22y+5$$

Factor

$$\left(2y+5\right)\left(4y+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8y^{2}+ay+by+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 positive, a and b are both positive. 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.

$$a=2 b=20$$

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

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

Factor out 2y in the first and 5 in the second group.

$$2y\left(4y+1\right)+5\left(4y+1\right)$$

Factor out common term 4y+1 by using distributive property.

$$\left(4y+1\right)\left(2y+5\right)$$

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.

$$8y^{2}+22y+5=0$$

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.

$$y=\frac{-22±\sqrt{22^{2}-4\times 8\times 5}}{2\times 8}$$

Square 22.

$$y=\frac{-22±\sqrt{484-4\times 8\times 5}}{2\times 8}$$

Multiply -4 times 8.

$$y=\frac{-22±\sqrt{484-32\times 5}}{2\times 8}$$

Multiply -32 times 5.

$$y=\frac{-22±\sqrt{484-160}}{2\times 8}$$

Add 484 to -160.

$$y=\frac{-22±\sqrt{324}}{2\times 8}$$

Take the square root of 324.

$$y=\frac{-22±18}{2\times 8}$$

Multiply 2 times 8.

$$y=\frac{-22±18}{16}$$

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

$$y=-\frac{4}{16}$$

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

$$y=-\frac{1}{4}$$

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

$$y=-\frac{40}{16}$$

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

$$y=-\frac{5}{2}$$

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

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

Simplify all the expressions of the form p-\left(-q\right) to p+q.

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

Add \frac{1}{4} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Add \frac{5}{2} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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

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

Multiply 4 times 2.

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

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

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

Evaluate

$$\left(2y+5\right)\left(4y+1\right)$$