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

To express the given quadratic equation \( x^2 + 6x + 5 = 0 \) as a square of a binomial, we need to identify whether it is a perfect square trinomial. A perfect square trinomial takes the form:

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

Let's see if we can rewrite \( x^2 + 6x + 5 \) in this form.

1. Identify the square term \( x^2 \). Here, \( a = x \).

2. The middle term is \( 6x \). This term should match \( 2ab \), where \( b \) is the number we are looking for.

3. We know \( 2ab = 6x \), so \( 2 \times x \times b = 6x \). Simplifying this, we get \( b = 3 \).

4. The square of \( b \) would be \( b^2 = 3^2 = 9 \).

Thus, the perfect square trinomial would be \( x^2 + 6x + 9 \), which can be expressed as \( (x + 3)^2 \).

However, since the original quadratic is \( x^2 + 6x + 5 \), and not \( x^2 + 6x + 9 \), it is **not** a perfect square trinomial. Instead, you can complete the square by adding and subtracting 4 within the equation:

\[
x^2 + 6x + 9 - 4 = 0
\]

This becomes:

\[
(x + 3)^2 - 4 = 0
\]

Therefore, the quadratic equation can be rewritten as:

\[
(x + 3)^2 = 4
\]

This step allows you to solve for \( x \):

\[
x + 3 = \pm 2
\]

So, \( x = -1 \) or \( x = -5 \).

Would you like more details or have any questions?

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Here are five related questions:

1. How do you complete the square for any quadratic equation?
2. Can you solve a quadratic equation using the quadratic formula?
3. What are the roots of the equation \( x^2 + 6x + 5 = 0 \)?
4. What does it mean if a trinomial is a perfect square?
5. How do you factorize a quadratic trinomial?

**Tip:** Completing the square is a powerful method not only for solving quadratic equations but also for transforming them into vertex form, which can be useful in graphing parabolas.

Learn how to express the quadratic equation x² + 6x + 5 = 0 as a square of a binomial with detailed step-by-step solutions. This problem covers completing the square, perfect square trinomials, and quadratic formulas, suitable for advanced high school students.

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