(x-1)^2=-4 How Many Real Solutions

Jasonbradley

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Jun 17, 2024

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Solving for Real Solutions of (x-1)^2 = -4

This equation presents a unique situation where we need to determine the number of real solutions, meaning solutions that are actual numbers on the number line.

Understanding the Problem

The equation (x-1)^2 = -4 involves a squared term, which means it represents a parabola. Here's the breakdown:

• Squaring: Squaring a number always results in a non-negative value (0 or positive).
• Negative Result: The equation is set equal to -4, a negative number.

This conflict indicates that there are no real solutions to the equation.

Why No Real Solutions?

Let's visualize:

1. The Parabola: The graph of (x-1)^2 is a parabola that opens upwards, with its vertex at (1,0).
2. Negative Value: The equation asks for the x-values where the parabola intersects the line y = -4.
3. No Intersection: Because the parabola is entirely above the x-axis, it never intersects the line y = -4.

Therefore, there are no real numbers that can satisfy the equation (x-1)^2 = -4.

Complex Solutions

While there are no real solutions, we can explore the possibility of complex solutions involving the imaginary unit "i", where i² = -1. This approach leads to solutions in the complex number system.