Final answer:
The solution to the system of the quadratic equation f(x) = 2x² + x - 4 and the linear equation g(x) = -2x + 5 is found by setting the functions equal to each other. After simplification, the correct solution, verified by substituting the values into the equations, is (1, 7).
Step-by-step explanation:
To solve for the solution to the system of equations involving the quadratic function f(x) = 2x² + x - 4 and the linear function g(x) = -2x + 5, you need to find the value of x where the two functions are equal. This essentially means solving for x in the equation 2x² + x - 4 = -2x + 5.
First, set the quadratic function equal to the linear function:
Next, move all terms to one side to set the equation to zero:
This is now a quadratic equation that can be solved either by factoring, completing the square, or using the quadratic formula. Since this equation does not factor neatly, we would use the quadratic formula or complete the square. In this case, the student provided options, so we can plug in the x-values from choices (a) through (d) to see which one satisfies the equation.
By testing each option:
- For (-2, 10): 2(-2)² + 3(-2) - 9 = 0 → 8 - 6 - 9 = 0, which is not true.
- For (-1, 7): 2(-1)² + 3(-1) - 9 = 0 → 2 - 3 - 9 = 0, which is not true.
- For (0, 5): 2(0)² + 3(0) - 9 = 0 → 0, which is not true.
- For (1, 7): 2(1)² + 3(1) - 9 = 0 → 2 + 3 - 9 = 0, which is true.
Therefore, the solution to the system of equations is (1, 7), which corresponds to answer (d).