Final answer:
The equation x² = -7x + 7 can be rearranged to the standard quadratic form x² + 7x - 7 = 0. Checking the discriminant, which is 77 and positive, reveals that there are two real solutions to this quadratic equation.
Step-by-step explanation:
To determine the number of real solutions for the equation x² = -7x + 7, we need to rearrange it into the standard quadratic form. By moving all the terms to one side, we get:
x² + 7x - 7 = 0
This equation is of the form ax² + bx + c = 0, where a = 1, b = 7, and c = -7. To find the solutions, we use the quadratic formula:
x = √[-b ± √(b² - 4ac)] / (2a).
Since we are looking for real solutions, we must check the discriminant (b² - 4ac). If the discriminant is positive, there are two real solutions; if it is zero, there is one real solution; and if it's negative, there are no real solutions.
For our equation:
Discriminant = b² - 4ac = (7)² - 4(1)(-7) = 49 + 28 = 77.
Since 77 is positive, there are two real solutions to the equation.