Analyzing and Factoring a Quadratic Trinomial: x^2 + 5x + 6

x^2 + 5x + 6

To analyze the expression, x^2 + 5x + 6, we can determine its characteristics and potentially factor it or find its roots

To analyze the expression, x^2 + 5x + 6, we can determine its characteristics and potentially factor it or find its roots.

1. Characteristic of the expression:
The expression is a quadratic trinomial, indicated by the highest power of x being 2 (x^2), and it is written in standard form (ax^2 + bx + c).

2. Finding the factors:
To factorize the expression, we need to find two binomials that, when multiplied, equal the quadratic trinomial. Let’s start by breaking down the quadratic trinomial into its factors.

First, consider the coefficient of x^2, which is 1. The possible factor pairs of 1 are (1, 1) and (-1, -1). We then look at the constant term, which is 6. The possible factor pairs of 6 are (1, 6), (-1, -6), (2, 3), and (-2, -3).

We need to find a combination of factors from the coefficient of x^2 and the constant term whose sum equals the coefficient of x, which is 5. In this case, the factor combination that satisfies this condition is (2, 3), as 2 + 3 = 5.

Therefore, the expression x^2 + 5x + 6 can be factored to (x + 2)(x + 3).

3. Finding the roots:
If we want to find the roots of the quadratic trinomial, we set it equal to zero and solve for x. Using the factored form from step 2, we have:

(x + 2)(x + 3) = 0

To satisfy the equation, either (x + 2) must equal zero, or (x + 3) must equal zero. This gives us two possible solutions:

x + 2 = 0 –> x = -2
x + 3 = 0 –> x = -3

Therefore, the roots of the quadratic trinomial x^2 + 5x + 6 are x = -2 and x = -3.

In summary, the expression x^2 + 5x + 6 can be factored to (x + 2)(x + 3), and its roots are x = -2 and x = -3.

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