Verified Answer :

  • Identify the quadratic trinomial as \(x^2 + 5x - 14\).
  • Find two numbers that multiply to \(-14\) and add to \(5\), which are \(-2\) and \(7\).
  • Express the quadratic in factored form as \((x - 2)(x + 7)\).
  • Identify that \((x - 2)\) is one of the factors, corresponding to option D.
  • The correct option is \(\boxed{D}\).

Explanation

  1. Analysis of the Quadratic Trinomial

The expression to factorize is a quadratic trinomial of the form \(x^2 + bx + c\), which is \(x^2 + 5x - 14\). To factorize this expression, we seek two integers \(m\) and \(n\) such that their product is \(c = -14\) and their sum is \(b = 5\). This is a standard procedure for factoring quadratic expressions with a leading coefficient of 1.

  1. Finding Factors of the Quadratic Expression

We examine the pairs of factors for \(-14\):

  • \(1\) and \(-14\): sum is \(1 - 14 = -13\)
  • \(-1\) and \(14\): sum is \(-1 + 14 = 13\)
  • \(2\) and \(-7\): sum is \(2 - 7 = -5\)
  • \(-2\) and \(7\): sum is \(-2 + 7 = 5\)

We see that the pair \(-2\) and \(7\) satisfies both conditions: \(-2 \times 7 = -14\) and \(-2 + 7 = 5\).

  1. Factoring and Selecting the Correct Option

Using the values \(m = -2\) and \(n = 7\), we can write the quadratic trinomial as the product of two binomials:
\[x^2 + 5x - 14 = (x - 2)(x + 7)\]
The two factors of the expression are \((x - 2)\) and \((x + 7)\). Comparing these with the provided options:

A\((x + 14)\)
B\((x + 2)\)
C\((x + 7)\)
D\((x - 2)\)

Both \((x + 7)\) and \((x - 2)\) are valid factors. Selecting option D, the correct factor is \((x - 2)\).

Examples

Factoring quadratic expressions is a fundamental skill used in physics and engineering to solve problems involving motion. For example, if the height of an object thrown into the air is modeled by the equation \(h(t) = -t^2 + 5t + 14\), you can find when it hits the ground by setting \(h(t) = 0\), which simplifies to finding the roots of the factored quadratic expression \(-(t-7)(t+2) = 0\). This allows you to determine the time \(t\) when the height is zero.

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