Question

Maribel factored $y^2-30y+81$as ${(y-9)}^2$. Was she right or wrong? How do you know?

Short answer

The factors maribel found are wrong and the correct factors are$\left(y-27\right)\left(y-3\right)$

Step-by-step solution

Step 1. Given Information

The given expression is$y^2-30y+81$

Step 2. Explanation

We will use perfect square trinomial pattern that is using the pattern ${(a-b)}^2$

The first and last terms of trinomial $y^2-30y+81$are perfect squares that is,

$$\begin{gathered} {\left(y\right)}^2=y^2=a \\ 81=9^2=b \end{gathered}$$

Now, we will check the middle term,

$$\begin{gathered} 2ab=2y\times 9 \\ =18y \end{gathered}$$

Here middle term is not equal to the given middle term. Thus, the binomial square pattern does not fits the trinomial.

Step 3. Calculation

The given trinomial is of the form $a{x}^2+bx+c$. By comparing it with the given expression, we get, $a=1,b=-30,c=81$

Now, we will use the middle term splitting method to find the correct factors.

$$\begin{gathered} y^2-30y+81=y^2-27y-3y+81 \\ =y(y-27)-3(y-27) \\ =(y-3)(y-27) \end{gathered}$$