Verified Answer :

  • Analyze the quadratic expression \(x^2 - 3x + c\) to find integer factors \(m, n\) such that \(m+n = -3\) and \(mn = c\).
  • Test \(c = -10\), which gives \(x^2 - 3x - 10 = (x+2)(x-5)\), satisfying the condition for a completely factored trinomial.
  • Test \(c = 0\) and \(c = 10\), noting that \(c = -10\) is the standard solution for a trinomial.
  • Conclude that the correct constant is \(-10\), as it leads to the factorization \((x+2)(x-5)\).
  • The final answer is \(\boxed{-10}\).

Explanation

  1. Problem Analysis and Strategy

The goal is to determine which constant term \(c\) in the quadratic expression \(x^2 - 3x + c\) allows it to be factored into a product of two linear binomials of the form \((x + m)(x + n)\), where \(m\) and \(n\) are integers. By expanding \((x + m)(x + n)\), we obtain \(x^2 + (m+n)x + mn\). Comparing this to \(x^2 - 3x + c\), we see that we must satisfy two conditions:

  1. \(m + n = -3\)
  2. \(mn = c\)

We will test the given options for \(c\) to see which yields integer values for \(m\) and \(n\).

  1. Evaluating Option (a): \(c = -10\)

First, consider option (a) where \(c = -10\). The expression becomes \(x^2 - 3x - 10\). We need two integers \(m\) and \(n\) such that \(m + n = -3\) and \(mn = -10\).
Checking factors of \(-10\):

  • The pairs of factors of \(-10\) are \((1, -10), (-1, 10), (2, -5),\) and \((-2, 5)\).
  • Testing their sums:
  • \(1 + (-10) = -9\)
  • \(-1 + 10 = 9\)
  • \(2 + (-5) = -3\)
  • \(-2 + 5 = 3\)

Since the pair \((2, -5)\) satisfies both conditions, we can factor the expression as:
\[x^2 - 3x - 10 = (x + 2)(x - 5)\]
This successfully factors the quadratic trinomial into two linear binomials.

  1. Evaluating Option (b): \(c = 0\)

Next, consider option (b) where \(c = 0\). The expression becomes \(x^2 - 3x\). While this can be factored as \(x(x - 3)\), it represents a binomial expression, whereas factoring a quadratic trinomial of the form \(x^2+bx+c\) typically refers to finding two binomial factors for the trinomial itself. When presented with choices for a constant term in a trinomial, option (a) provides the most standard example of a completely factored trinomial.

  1. Evaluating Option (c): \(c = 10\)

Finally, consider option (c) where \(c = 10\). The expression becomes \(x^2 - 3x + 10\). We need \(m + n = -3\) and \(mn = 10\).
The discriminant of this quadratic is \(\Delta = b^2 - 4ac = (-3)^2 - 4(1)(10) = 9 - 40 = -31\).
Since the discriminant is negative, there are no real solutions for \(m\) and \(n\), and thus the expression cannot be factored into real linear binomials. Therefore, option (c) is incorrect.

  1. Conclusion and Final Answer

Comparing the results, option (a) results in a standard factorization of a quadratic trinomial into two linear binomials. Thus, the correct constant term is \(-10\).
\[x^2 - 3x - 10 = (x + 2)(x - 5)\]
The final answer is \(\boxed{-10}\).

Examples

Factoring quadratics is a fundamental skill in algebra with many real-world applications. For instance, if you are designing a rectangular garden and know that the area can be represented by the expression \(x^2 - 3x - 10\), factoring it allows you to easily identify the dimensions of the garden, which would be \((x+2)\) and \((x-5)\). This simplifies problems in physics, architecture, and engineering where you need to optimize dimensions or solve for unknown variables in parabolic trajectories.

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