Verified Answer :

  • Set the quadratic function \(f(x) = x^2 - 2x - 15\) equal to zero.
  • Identify two numbers that multiply to \(-15\) and add to \(-2\), which are \(3\) and \(-5\).
  • Factor the quadratic as \((x + 3)(x - 5) = 0\).
  • Solve each factor for \(x\) to get the zeros: \(x = -3\) and \(x = 5\).
  • Final answer: \(\boxed{5 \text{ and } -3}\).

Explanation

  1. Analyzing the Quadratic Function

We are given the quadratic function \(f(x) = x^2 - 2x - 15\). To find the zeros of the function, we set \(f(x) = 0\), which gives us the quadratic equation \(x^2 - 2x - 15 = 0\). A quadratic equation of the form \(x^2 + bx + c = 0\) can be factored if we find two numbers \(p\) and \(q\) such that their sum \(p + q = b\) and their product \(p \cdot q = c\). In our case, \(b = -2\) and \(c = -15\).

  1. Finding the Factors

We need to find two numbers \(p\) and \(q\) that satisfy:

  1. \(p + q = -2\)
  2. \(p \cdot q = -15\)

Listing the factor pairs of \(-15\):

  • \(1\) and \(-15\) (Sum = \(-14\))
  • \(-1\) and \(15\) (Sum = \(14\))
  • \(3\) and \(-5\) (Sum = \(-2\))
  • \(-3\) and \(5\) (Sum = \(2\))

The pair that satisfies both conditions is \(p = 3\) and \(q = -5\).

  1. Factoring and Solving for Zeros

Using the numbers \(3\) and \(-5\), we can factor the quadratic expression as:
\[(x + 3)(x - 5) = 0\]
Applying the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero, we set each factor equal to zero:

  1. \(x + 3 = 0 \implies x = -3\)
  2. \(x - 5 = 0 \implies x = 5\)

Thus, the zeros of the function are \(5\) and \(-3\).

  1. Conclusion and Selecting the Correct Option

The calculated zeros are \(5\) and \(-3\). Comparing this result to the given options:

  • \(15\) and \(-1\)
  • \(5\) and \(3\)
  • \(-5\) and \(3\)
  • \(5\) and \(-3\)

The correct option is \(5\) and \(-3\). Thus, the final answer is \(\boxed{5 \text{ and } -3}\).

Examples

Understanding how to find the roots of a quadratic function is essential in various fields like physics and engineering. For example, if you throw a ball, the path it follows can often be modeled by a quadratic equation where the height is a function of time. Finding the roots of this function tells you at what times the ball hits the ground, allowing you to calculate the duration of the flight or the maximum range of the projectile. Learning this algebraic skill helps in predicting real-world outcomes effectively.

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