Solve 1/3y^2-4y+9

$$\frac { 1 } { 3 } y ^ { 2 } - 4 y + 9$$

Factor

$$\frac{\left(y-9\right)\left(y-3\right)}{3}$$

Solution Steps

Factor out \frac{1}{3}.

$$\frac{y^{2}-12y+27}{3}$$

Consider y^{2}-12y+27. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+27. To find a and b, set up a system to be solved.

$$a+b=-12 ab=1\times 27=27$$

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 27.

$$-1,-27 -3,-9$$

Calculate the sum for each pair.

$$-1-27=-28 -3-9=-12$$

The solution is the pair that gives sum -12.

$$a=-9 b=-3$$

Rewrite y^{2}-12y+27 as \left(y^{2}-9y\right)+\left(-3y+27\right).

$$\left(y^{2}-9y\right)+\left(-3y+27\right)$$

Factor out y in the first and -3 in the second group.

$$y\left(y-9\right)-3\left(y-9\right)$$

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

$$\left(y-9\right)\left(y-3\right)$$

Rewrite the complete factored expression.

$$\frac{\left(y-9\right)\left(y-3\right)}{3}$$

Evaluate

$$\frac{\left(y-9\right)\left(y-3\right)}{3}$$