- Identify the quadratic equation components \(a=1, b=-8, c=-20\).
- Factor the expression as \((x - 10)(x + 2) = 0\) to find the roots.
- Verify the roots using the quadratic formula, resulting in \(x = 10\) and \(x = -2\).
- Select the matching option, which is D. \(\boxed{\text{D}}\).
Explanation
- Analysis of the Quadratic Equation
We are tasked with solving 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 and identify the correct option from the given choices.
- Factoring the Quadratic Equation
To solve \(x^2 - 8x - 20 = 0\), we can use the factoring method. We look for two numbers that multiply to the constant term \(c = -20\) and add up to the coefficient \(b = -8\). Testing the factor pairs of \(-20\):
- \(1 \times -20 = -20\), sum = \(-19\)
- \(-1 \times 20 = -20\), sum = \(19\)
- \(2 \times -10 = -20\), sum = \(-8\)
The pair that works is \(2\) and \(-10\). Thus, the equation can be written as \((x - 10)(x + 2) = 0\). Setting each factor equal to zero gives us the roots:
- \(x - 10 = 0 \implies x = 10\)
- \(x + 2 = 0 \implies x = -2\)
- Verifying with the Quadratic Formula
Alternatively, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
For our equation, \(a=1, b=-8, c=-20\).
The discriminant is \(D = b^2 - 4ac = (-8)^2 - 4(1)(-20) = 64 + 80 = 144\).
The square root of the discriminant is \(\sqrt{144} = 12\).
Now, substitute these into the formula:
\(x = \frac{-(-8) \pm 12}{2(1)} = \frac{8 \pm 12}{2}\).
Calculating the two solutions:
- \(x_1 = \frac{8 + 12}{2} = \frac{20}{2} = 10\)
- \(x_2 = \frac{8 - 12}{2} = \frac{-4}{2} = -2\)
Both methods confirm the roots are \(-2\) and \(10\). Comparing these results with the given options, we see that option D is the correct one.
Examples
Quadratic equations are used in many real-world scenarios, such as modeling the trajectory of a projectile. If you throw a ball, its height \(h\) at time \(t\) can often be represented by a quadratic equation, such as \(h(t) = -5t^2 + 10t + 2\). By solving for when \(h(t) = 0\), you can determine when the ball hits the ground. Understanding how to find these roots allows us to predict physical outcomes accurately.