Verified Answer :

  • The given quadratic equation is \(x^2 - 8x - 20 = 0\).
  • Factoring the expression yields \((x - 10)(x + 2) = 0\).
  • Solving for \(x\) gives the roots \(x = 10\) and \(x = -2\).
  • The correct option is D, \(\boxed{-2 \text{ and } 10}\).

Explanation

  1. Analyzing the Equation

We are given the quadratic equation \(x^2 - 8x - 20 = 0\). This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -8\), and \(c = -20\). Our goal is to find the values of \(x\) that satisfy this equation. We can approach this by factoring the quadratic expression or by using the quadratic formula. Given the coefficients, factoring is an efficient method.

  1. Factoring the Quadratic Expression

To factor the quadratic \(x^2 - 8x - 20 = 0\), we look for two numbers that multiply to the constant term \(c = -20\) and add up to the coefficient of the linear term \(b = -8\).
Checking factors of -20:

  • \(-20 \times 1 = -20\), sum = -19
  • \(-10 \times 2 = -20\), sum = -8
  • \(-5 \times 4 = -20\), sum = -1

The pair \(-10\) and \(2\) works perfectly. Thus, we can rewrite the quadratic equation as \((x - 10)(x + 2) = 0\).

  1. Conclusion and Solution Selection

Now we set each factor equal to zero:
\(x - 10 = 0 \implies x = 10\)
\(x + 2 = 0 \implies x = -2\)
Thus, the solutions to the equation are \(x = 10\) and \(x = -2\). Comparing these with the provided options, we see that Option D lists these exact values. We can verify this by substituting \(x = 10\): \(10^2 - 8(10) - 20 = 100 - 80 - 20 = 0\), and for \(x = -2\): \((-2)^2 - 8(-2) - 20 = 4 + 16 - 20 = 0\). Both satisfy the original equation.

Examples

Quadratic equations like \(x^2 - 8x - 20 = 0\) often appear in physics when calculating the position of an object moving with constant acceleration. For instance, if you are modeling the distance an object travels over time to find when it hits a target, you might end up with an equation representing the time intervals. Solving these equations helps you accurately predict movement, plan logistics, or optimize designs in various engineering projects.

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