Final answer:
The correct equation Ryan started with is option d, y = x² + 8x - 2. After using the quadratic formula and confirming with x = -1, this equation yields a consistent result. Quadratic equations generally have two solutions unless the discriminant is zero.
Step-by-step explanation:
To determine which quadratic equation Ryan started with and how many solutions he should find, we must look for an equation in standard form, ax²+bx+c = 0, that would yield x = -1 as a solution when applying the quadratic formula. By substituting x = -1 into each equation option, we can confirm which one has this value as a solution. Let's evaluate option a: if x = -1, then y = -2(-1)² + 8(-1) - 4, which simplifies to y = -2 - 8 - 4 = -14, so this is not the equation Ryan used, as he got x = -1 as a solution, not y = -14. Option b is a linear equation, y = -2x + 8, and would only have one solution, but the given solution x = -1 does not satisfy it. Option c seems to be a typo, but even assuming it's y = x² - 8x - 2, substituting x = -1 gives y = (-1)² - 8(-1) - 2 = 1 + 8 - 2 = 7, which is again incorrect. Therefore, the correct equation is option d, which is y = x² + 8x - 2. Substituting x = -1 gives us y = (-1)² + 8(-1) - 2 = 1 - 8 - 2 = -9, which matches the result of the quadratic formula. Since the equation is quadratic, two solutions are generally expected unless the discriminant is zero, resulting in one real repeated solution.