Verified Answer :

  • Use the AC method to find two numbers that multiply to \(ac = 30\) and add to \(b = 13\), which are \(3\) and \(10\).
  • Rewrite the middle term \(13x\) as \(10x + 3x\) to get \(5x^2 + 10x + 3x + 6\).
  • Group the terms as \((5x^2 + 10x) + (3x + 6)\) and factor out the common terms: \(5x(x + 2) + 3(x + 2)\).
  • Factor out the common binomial \((x + 2)\) to reach the final form \(\boxed{(5x + 3)(x + 2)}\).

Explanation

  1. Problem Analysis

The goal is to factor the quadratic trinomial \(5x^2 + 13x + 6\). This expression is in the standard form \(ax^2 + bx + c\), where the coefficients are \(a = 5\), \(b = 13\), and \(c = 6\). To factor this trinomial, we can use the "AC method," which involves finding two numbers that multiply to the product of \(a\) and \(c\) (which is \(5 \times 6 = 30\)) and add up to the middle coefficient \(b = 13\).

  1. Finding the Factors Using the AC Method

We need to find two numbers \(m\) and \(n\) such that:
1) \(m \times n = a \times c = 5 \times 6 = 30\)
2) \(m + n = b = 13\)

Looking at the factors of \(30\):

  • \(1 \times 30 = 30\), and \(1 + 30 = 31\)
  • \(2 \times 15 = 30\), and \(2 + 15 = 17\)
  • \(3 \times 10 = 30\), and \(3 + 10 = 13\)

The numbers we need are \(3\) and \(10\).

  1. Grouping and Factoring

Now, rewrite the middle term \(13x\) using the two numbers found: \(13x = 10x + 3x\). The expression becomes:
\[5x^2 + 10x + 3x + 6\]
Next, group the terms to facilitate factoring:
\[(5x^2 + 10x) + (3x + 6)\]
Factor out the greatest common factor from each group:

  • From \((5x^2 + 10x)\), factor out \(5x\), which gives \(5x(x + 2)\).
  • From \((3x + 6)\), factor out \(3\), which gives \(3(x + 2)\).

The expression is now:
\[5x(x + 2) + 3(x + 2)\]

  1. Conclusion and Verification

Since the binomial \((x + 2)\) is a common factor, we can factor it out:
\[(5x + 3)(x + 2)\]
We can verify the result by multiplying the binomials:
\[(5x + 3)(x + 2) = 5x^2 + 10x + 3x + 6 = 5x^2 + 13x + 6\]
This confirms the factorization is correct. The factored form is \((5x + 3)(x + 2)\).

Examples

Factoring quadratic trinomials is a fundamental skill used in physics and engineering to solve problems involving motion. For example, if the height of a projectile is given by a quadratic expression such as \(-5t^2 + 13t + 6\), factoring it can help find the exact time \(t\) when the projectile hits the ground (when the expression equals zero). Mastering this skill allows for quick analysis of such physical systems by identifying the roots of the quadratic function.

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