perfect square trinomial into square of a binomial x²+6x+5=0

The expression you provided, \( x^2 + 6x + 5 = 0 \), is a quadratic equation. However, it is not a perfect square trinomial because the constant term doesn't allow it to factor into a perfect square binomial. 

A perfect square trinomial has the form:

\[
a^2 + 2ab + b^2 = (a + b)^2
\]

To determine whether \( x^2 + 6x + 5 \) is a perfect square trinomial, we compare it with the form above:

1. The first term \( x^2 \) suggests \( a = x \).
2. The second term \( 6x \) suggests \( 2ab = 6x \), so \( b = 3 \) if \( a = x \).
3. The third term should be \( b^2 = 9 \) to match a perfect square, but instead, we have 5.

Since the third term is 5, this trinomial isn't a perfect square. However, you can factor the quadratic equation using other methods, such as factoring or the quadratic formula.

### Factoring the Quadratic:

Let's factor \( x^2 + 6x + 5 \):

\[
x^2 + 6x + 5 = (x + 1)(x + 5)
\]

So the equation can be rewritten as:

\[
(x + 1)(x + 5) = 0
\]

### Solving the Equation:

Set each factor to zero:

\[
x + 1 = 0 \quad \text{or} \quad x + 5 = 0
\]

\[
x = -1 \quad \text{or} \quad x = -5
\]

Thus, the solutions are \( x = -1 \) and \( x = -5 \).

Would you like more details on this, or do you have any further questions?

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Here are five related questions:

1. How do you identify a perfect square trinomial?
2. Can you always factor a quadratic equation?
3. What is the quadratic formula, and when should you use it?
4. How do you complete the square for any quadratic equation?
5. What are the roots of a quadratic equation?

**Tip:** Always check the discriminant (\( b^2 - 4ac \)) to determine the nature of the roots of a quadratic equation before attempting to factor it.

Explore how to solve the quadratic equation x² + 6x + 5 = 0 by factoring. This detailed analysis covers key algebraic concepts such as factoring quadratic trinomials and using the quadratic formula. Suitable for Grades 9-12.

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