Certainly! We need to factor the quadratic expression [tex]\( y^2 - 10y + 25 \)[/tex].
### Step 1: Identify and Arrange Terms
The given expression is [tex]\( y^2 - 10y + 25 \)[/tex].
### Step 2: Look for a Pattern
We can see if this expression fits the pattern of a perfect square trinomial. A perfect square trinomial is of the form:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
or
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2. \][/tex]
### Step 3: Compare with Perfect Square Trinomial
Let's compare [tex]\( y^2 - 10y + 25 \)[/tex] with [tex]\( (a - b)^2 \)[/tex]:
- Here, [tex]\( a = y \)[/tex].
- We need to find [tex]\( b \)[/tex] such that [tex]\( y^2 - 10y + 25 \)[/tex] fits the pattern [tex]\( y^2 - 2by + b^2 \)[/tex].
### Step 4: Determine [tex]\( b \)[/tex]
From the given expression:
- The middle term is [tex]\(-10y\)[/tex]. Therefore, [tex]\(-2by = -10y\)[/tex].
We solve for [tex]\( b \)[/tex]:
[tex]\[ -2b = -10 \implies b = 5. \][/tex]
### Step 5: Confirm with the Last Term
We need to check if [tex]\( 5^2 \)[/tex] matches the constant term in the given quadratic expression:
[tex]\[ 5^2 = 25. \][/tex]
The constant term in the given expression is 25, which matches [tex]\( 5^2 \)[/tex].
### Step 6: Write the Factored Form
Now that we have identified [tex]\( b \)[/tex], we can write the expression in its factored form:
[tex]\[ (y - 5)^2. \][/tex]
So, the quadratic expression [tex]\( y^2 - 10y + 25 \)[/tex] factors to:
[tex]\[ (y - 5)^2. \][/tex]
This factored form confirms the expression is indeed a perfect square trinomial. Therefore, the fully factored form is:
[tex]\[ (y - 5)^2. \][/tex]
