Verified Answer :

  • Factor the quadratic expression to find the roots: \(x^2 + 3x - 4 = (x + 4)(x - 1) > 0\).
  • Determine that the inequality holds when \(x < -4\) or \(x > 1\).
  • Test the candidates: \(-6 < -4\) and \(5 > 1\) satisfy the inequality, while \(-2\) and \(0\) do not.
  • The two correct values are \(\boxed{-6, 5}\).

Explanation

  1. Finding the Roots of the Quadratic Expression

The inequality to evaluate is \(x^2 + 3x - 4 > 0\). To determine the set of values for which this inequality holds, we first identify the roots of the corresponding quadratic equation \(x^2 + 3x - 4 = 0\). Factoring the quadratic expression, we look for two numbers that multiply to \(-4\) and add to \(3\). These numbers are \(4\) and \(-1\). Thus, the equation can be rewritten as \((x + 4)(x - 1) = 0\). The roots of this equation are \(x = -4\) and \(x = 1\).

  1. Determining the Solution Interval

Since the quadratic \(x^2 + 3x - 4\) represents a parabola that opens upwards (because the coefficient of \(x^2\) is positive), the expression is greater than zero outside the interval between the two roots. Therefore, the inequality \(x^2 + 3x - 4 > 0\) is satisfied when \(x < -4\) or \(x > 1\). Any value outside the range \([-4, 1]\) will satisfy the inequality.

  1. Evaluating the Candidate Values

We now check each of the given candidate values to see if they satisfy the condition \(x < -4\) or \(x > 1\):

  • For \(x = -2\): \(-2\) is within the interval \([-4, 1]\), so it does not satisfy the inequality.
  • For \(x = 0\): \(0\) is within the interval \([-4, 1]\), so it does not satisfy the inequality.
  • For \(x = -6\): \(-6 < -4\), so this value satisfies the inequality.
  • For \(x = 5\): \(5 > 1\), so this value satisfies the inequality.

Thus, the candidates that satisfy the inequality are \(-6\) and \(5\).

Examples

Quadratic inequalities appear in many real-life optimization problems. For instance, if you are designing a product and the profit \(P\) from selling \(x\) units is modeled by \(P(x) = -x^2 + 10x - 16\), you might want to find the range of production units \(x\) that ensures the company makes a profit (i.e., \(P(x) > 0\)). Solving this quadratic inequality helps determine the optimal range for production to guarantee that the business remains profitable.

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