Use the graphing calculator to graph the quadratic function y=2x2−6x−8.
The Correct Answer and Explanation is:
The correct options are -1 and 4.
Explanation:
The problem asks for the solutions to the quadratic equation 0 = 2x² – 6x – 8. Finding the solutions, or roots, of this equation is equivalent to finding the x-intercepts of the related quadratic function y = 2x² – 6x – 8. The x-intercepts are the points on the graph where the function crosses the x-axis, which occurs when the y-value is zero.
The prompt suggests using a graphing calculator. If you were to graph the function y = 2x² – 6x – 8, you would see a parabola that opens upwards. The points where this parabola intersects the horizontal x-axis are the solutions. The graph would cross the x-axis at x = -1 and x = 4. Therefore, these are the two solutions to the equation.
Even without a graphing calculator, we can verify the solutions algebraically. The most direct method for a multiple-choice question is to substitute the given options into the equation and see which ones make the statement true.
Let’s test the correct options:
- Test x = -1:
- 2(-1)² – 6(-1) – 8
- 2(1) – (-6) – 8
- 2 + 6 – 8
- 8 – 8 = 0
Since 0 = 0, the equation holds true. Thus, -1 is a solution.
- Test x = 4:
- 2(4)² – 6(4) – 8
- 2(16) – 24 – 8
- 32 – 24 – 8
- 8 – 8 = 0
Since 0 = 0, the equation also holds true. Thus, 4 is a solution.
Another algebraic method is factoring. First, we can simplify the equation by dividing all terms by their greatest common factor, which is 2:
(2x² – 6x – 8) / 2 = 0 / 2
x² – 3x – 4 = 0
Next, we look for two numbers that multiply to -4 and add to -3. These numbers are -4 and 1. We can then write the equation in factored form:
(x – 4)(x + 1) = 0
According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.
- x – 4 = 0 → x = 4
- x + 1 = 0 → x = -1
Both algebraic methods confirm that the two solutions are -1 and 4
