
If a quadratic has the same root twice, then the standard form will be in the form 
x 
2
 +2r+r 
2
  this is called a "perfect square trinomial" and the factors will be 
(x−r)(x−r) or 
(x−r) 
2
  use this pattern to factor the following trinomial into a binomial squared. 

x 
2
 −10x+25

We are given the quadratic \( x^2 - 10x + 25 \). To factor it into a binomial squared, we need to check if it fits the pattern of a **perfect square trinomial**, which is of the form:

\[
x^2 + 2rx + r^2 = (x - r)^2
\]

### Step-by-step factoring process:

1. Compare the given trinomial \( x^2 - 10x + 25 \) with the general form \( x^2 + 2rx + r^2 \).
   
   - The coefficient of \( x \) is \( -10 \), and this corresponds to \( 2r \) in the general form. So, we have:
   \[
   2r = -10 \implies r = -5
   \]

2. Check if the constant term \( 25 \) matches \( r^2 \):
   \[
   r^2 = (-5)^2 = 25
   \]
   This confirms that the trinomial is a perfect square trinomial.

3. Now, use the value of \( r = -5 \) to write the trinomial as a binomial squared:
   \[
   x^2 - 10x + 25 = (x - 5)^2
   \]

Thus, the factorization of \( x^2 - 10x + 25 \) is:

\[
(x - 5)^2
\]

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do you determine if a quadratic is a perfect square trinomial?
2. Can every quadratic trinomial be factored into a binomial squared?
3. What is the relationship between the discriminant and having repeated roots?
4. How do you factor a quadratic when it isn't a perfect square trinomial?
5. How does completing the square relate to perfect square trinomials?

**Tip:** A perfect square trinomial always has its middle term as double the product of the square roots of the first and last terms.

If a quadratic has the same root twice, then the standard form will be in the form x^2 + 2r + r^2. This is called a 'perfect square trinomial' and the factors will be (x−r)(x−r) or (x−r)^2. Use this pattern to factor the following trinomial into a binomial squared: x^2 −10x+25

Learn how to factor the perfect square trinomial x^2 - 10x + 25 into (x - 5)^2. This problem explains the pattern of perfect square trinomials, showing how to recognize and factor them using algebraic methods. Suitable for Grades 8-10 students studying quadratic equations and factoring techniques.

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