Question

StartFraction y squared minus 3 y plus 2 Over y squared plus 5 y minus 6 EndFraction

Answer 1

To simplify the expression

\[
\frac{y^2 - 3y + 2}{y^2 + 5y - 6},
\]

we can start by factoring both the numerator and the denominator.

**Step 1: Factor the numerator \(y^2 - 3y + 2\)**

We need to find two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(y\)). The numbers \(-1\) and \(-2\) satisfy these conditions.

Thus, we can factor the numerator:

\[
y^2 - 3y + 2 = (y - 1)(y - 2).
\]

**Step 2: Factor the denominator \(y^2 + 5y - 6\)**

Next, we look for two numbers that multiply to \(-6\) and add to \(5\). The numbers \(6\) and \(-1\) fit these requirements.

So, we factor the denominator:

\[
y^2 + 5y - 6 = (y + 6)(y - 1).
\]

**Step 3: Rewrite the expression**

Now, we substitute the factored forms back into the expression:

\[
\frac{(y - 1)(y - 2)}{(y + 6)(y - 1)}.
\]

**Step 4: Simplify the expression**

We notice that \((y - 1)\) appears in both the numerator and the denominator, and we can cancel it out (provided \(y \neq 1\)):

\[
\frac{(y - 2)}{(y + 6)} \quad \text{for } y \neq 1.
\]

**Final Result:**

Thus, the simplified form of the expression is:

\[
\frac{y - 2}{y + 6} \quad \text{for } y \neq 1.
\]