Verified Answer :

  • Identify coefficients \(a=1, b=7, c=12\).
  • Find two numbers whose product is \(12\) and sum is \(7\), which are \(3\) and \(4\).
  • Write the expression as the product of two binomials \((x+3)(x+4)\).
  • Verify the result by expanding the binomials: \(\boxed{(x+3)(x+4)}\).

Explanation

  1. Analyzing the Quadratic Expression

To factor the quadratic expression \(x^2 + 7x + 12\), we need to represent it in the form \((x + m)(x + n)\). This is equivalent to expanding \((x + m)(x + n) = x^2 + (m+n)x + mn\). By comparing this to the given expression \(x^2 + 7x + 12\), we see that we need to find two numbers \(m\) and \(n\) such that their sum \(m + n = 7\) and their product \(mn = 12\).

  1. Finding the Factors

We look for factors of the constant term \(12\) that add up to the coefficient of the middle term, which is \(7\). The factor pairs of \(12\) are:

  • \(1\) and \(12\) (sum: \(1 + 12 = 13\))
  • \(2\) and \(6\) (sum: \(2 + 6 = 8\))
  • \(3\) and \(4\) (sum: \(3 + 4 = 7\))

The pair \((3, 4)\) satisfies both conditions because their product is \(3 \times 4 = 12\) and their sum is \(3 + 4 = 7\).

  1. Writing the Factored Form and Verification

Now that we have identified the two numbers \(m=3\) and \(n=4\), we can substitute them into the factored form \((x + m)(x + n)\). Therefore, the expression \(x^2 + 7x + 12\) can be written as \((x + 3)(x + 4)\). We can verify this by expanding the product:
\((x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12\).
The factored expression is correct.

Examples

Factoring quadratic expressions is a fundamental skill in algebra used to solve equations and simplify rational expressions. For example, if you are designing a rectangular garden and you know its area is represented by \(x^2 + 7x + 12\) square meters, factoring this expression helps you determine the dimensions of the garden in terms of \(x\), which could be \((x+3)\) meters by \((x+4)\) meters. This is a common application in architecture, landscape design, and engineering where physical dimensions need to be expressed as products of simpler binomials.

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