Solve 9y^2+13y+4

$$9 { y }^{ 2 } +13y+4$$

Factor

$$\left(y+1\right)\left(9y+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9y^{2}+ay+by+4. To find a and b, set up a system to be solved.

$$a+b=13 ab=9\times 4=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 13.

$$a=4 b=9$$

Rewrite 9y^{2}+13y+4 as \left(9y^{2}+4y\right)+\left(9y+4\right).

$$\left(9y^{2}+4y\right)+\left(9y+4\right)$$

Factor out y in 9y^{2}+4y.

$$y\left(9y+4\right)+9y+4$$

Factor out common term 9y+4 by using distributive property.

$$\left(9y+4\right)\left(y+1\right)$$

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.

$$9y^{2}+13y+4=0$$

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.

$$y=\frac{-13±\sqrt{13^{2}-4\times 9\times 4}}{2\times 9}$$

Square 13.

$$y=\frac{-13±\sqrt{169-4\times 9\times 4}}{2\times 9}$$

Multiply -4 times 9.

$$y=\frac{-13±\sqrt{169-36\times 4}}{2\times 9}$$

Multiply -36 times 4.

$$y=\frac{-13±\sqrt{169-144}}{2\times 9}$$

Add 169 to -144.

$$y=\frac{-13±\sqrt{25}}{2\times 9}$$

Take the square root of 25.

$$y=\frac{-13±5}{2\times 9}$$

Multiply 2 times 9.

$$y=\frac{-13±5}{18}$$

Now solve the equation y=\frac{-13±5}{18} when ± is plus. Add -13 to 5.

$$y=-\frac{8}{18}$$

Reduce the fraction \frac{-8}{18} to lowest terms by extracting and canceling out 2.

$$y=-\frac{4}{9}$$

Now solve the equation y=\frac{-13±5}{18} when ± is minus. Subtract 5 from -13.

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

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{4}{9} for x_{1} and -1 for x_{2}.

$$9y^{2}+13y+4=9\left(y-\left(-\frac{4}{9}\right)\right)\left(y-\left(-1\right)\right)$$

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

$$9y^{2}+13y+4=9\left(y+\frac{4}{9}\right)\left(y+1\right)$$

Add \frac{4}{9} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9y^{2}+13y+4=9\times \frac{9y+4}{9}\left(y+1\right)$$

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

$$9y^{2}+13y+4=\left(9y+4\right)\left(y+1\right)$$

Evaluate

$$\left(y+1\right)\left(9y+4\right)$$