Final answer:
The discriminant of the equation 5x²-49x+421 = x²+7x-3, after rearranging to 4x² - 56x + 424 = 0, is found to be -3648, indicating there are no real solutions and two complex solutions.
Step-by-step explanation:
First, we need to combine like terms in the given quadratic equation 5x²-49x+421 = x²+7x-3 to bring it to the standard form ax² + bx + c = 0. Subtract x² from both sides to get 4x², subtract 7x from both sides to get -56x, and add 3 to both sides to get 424, resulting in the standard form 4x² - 56x + 424 = 0.
We find the discriminant by using the formula Δ = b² - 4ac, where a, b, and c are coefficients from the standard form of the quadratic equation. For our equation, a = 4, b = -56, and c = 424, thus the discriminant is Δ = (-56)² - 4(4)(424).
Calculating the discriminant yields Δ = 3136 - 6784 = -3648. Since the discriminant is negative, this indicates the equation has no real solutions; there are two complex solutions.