Find the number of real solutions to the following equations: 7) x + 3 = (x – 5)^2 8) x^4 = x – 10 9) x^3 = 5 – x

To find the number of real solutions to the equation x + 3 = (x – 5)^2, we can rewrite the equation by expanding the right side and then setting it equal to zero to find the roots.

Expanded, the equation is: x + 3 = x^2 - 10x + 25 Rearrange the terms by bringing them all to one side to obtain a quadratic equation: x^2 - 10x + 25 - x - 3 = 0 Combine like terms: x^2 - 11x + 22 = 0

Now we can use the discriminant, b^2 - 4ac, from the quadratic formula to determine the nature of its solutions.

Here, a = 1, b = -11, and c = 22, so the discriminant is: (-11)^2 - 4(1)(22) = 121 - 88 = 33

Since the discriminant is positive, the equation x^2 - 11x + 22 = 0 has two distinct real solutions.

8) For the equation x^4 = x - 10, to find the real solutions, this requires factoring the equation or other numerical methods, as it is not a simple polynomial. However, we can investigate the number of real solutions using graphing or calculus methods like the Intermediate Value Theorem.

9) For the equation x^3 = 5 - x, we would look for points where the functions f(x) = x^3 and g(x) = 5 - x intersect. This equation is not readily factorable, but we can again use graphing or calculus tools to investigate the real solutions. By analyzing the behavior of a cubic function and a linear function, it suggests that there will be at least one real solution because a cubic function is continuous and will intersect the line at least once. To find the exact number of real solutions, we might need to do graphical analysis or apply more sophisticated methods.