Verified Answer :

  • Rewrite the inequality \(-x^2 + 2x + 35 > 0\) as \(x^2 - 2x - 35 < 0\) by multiplying by \(-1\).
  • Factor the quadratic expression into \((x - 7)(x + 5) < 0\).
  • Identify the roots \(x = 7\) and \(x = -5\).
  • Determine that the expression is negative between the roots, resulting in the solution \(\boxed{-5 < x < 7}\).

Explanation

  1. Problem Analysis

The problem asks for the interval of values for \(x\) that satisfy the quadratic inequality \(-x^2 + 2x + 35 > 0\). This is a quadratic expression of the form \(ax^2 + bx + c\), where \(a = -1\), \(b = 2\), and \(c = 35\). Since the coefficient of \(x^2\) is negative (\(a = -1\)), the graph of the function \(f(x) = -x^2 + 2x + 35\) is a parabola that opens downward. We are looking for the region where this parabola is strictly above the \(x\)-axis.

  1. Finding the Roots

To simplify the inequality, we multiply both sides by \(-1\). Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. Thus, the inequality becomes:
\[x^2 - 2x - 35 < 0\]
We now need to find the roots of the corresponding quadratic equation \(x^2 - 2x - 35 = 0\) to determine the boundaries of the interval. We can factor this quadratic by looking for two numbers that multiply to \(-35\) and add to \(-2\). Those numbers are \(-7\) and \(5\). Therefore, we can write the quadratic as:
\[(x - 7)(x + 5) = 0\]
The roots are \(x = 7\) and \(x = -5\).

  1. Determining the Solution Interval

The expression \((x - 7)(x + 5) < 0\) is negative when one factor is positive and the other is negative. Testing values in the intervals defined by the roots \(x = -5\) and \(x = 7\):

  • For \(x < -5\) (e.g., \(x = -6\)): \((-6 - 7)(-6 + 5) = (-13)(-1) = 13 > 0\).
  • For \(-5 < x < 7\) (e.g., \(x = 0\)): \((0 - 7)(0 + 5) = (-7)(5) = -35 < 0\).
  • For \(x > 7\) (e.g., \(x = 8\)): \((8 - 7)(8 + 5) = (1)(13) = 13 > 0\).

The inequality \(x^2 - 2x - 35 < 0\) is satisfied when \(-5 < x < 7\). This matches the original inequality \(-x^2 + 2x + 35 > 0\).

  1. Conclusion

Based on our analysis, the values of \(x\) that satisfy the inequality \(-x^2 + 2x + 35 > 0\) are those in the interval \(-5 < x < 7\). This corresponds to the third option provided in the problem description. Therefore, the correct interval is \(-5 < x < 7\).

Examples

Quadratic inequalities are used in real-life applications like physics and engineering to determine safe operational ranges. For instance, if you are throwing a ball and the height \(h\) at time \(t\) is modeled by \(h(t) = -t^2 + 2t + 35\), you might need to find when the ball is above a certain height, such as ground level (\(h > 0\)). By solving \(-t^2 + 2t + 35 > 0\), you can identify the time interval during which the ball remains in the air before hitting the ground.

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