(x-5)^2=-9

Jasonbradley

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Jun 17, 2024

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Solving the Equation (x - 5)² = -9

This equation presents a unique challenge because it involves squaring a binomial and setting it equal to a negative number. Let's break down the steps to find the solutions:

Understanding the Problem

The square of any real number is always non-negative (zero or positive). This means that there is no real number that, when squared, will result in -9. However, we can explore solutions in the realm of complex numbers.

Complex Numbers

Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).

Solving the Equation

1. Isolate the squared term: We already have the squared term isolated: (x - 5)² = -9

2. Take the square root of both sides: √[(x - 5)²] = ±√(-9)

3. Simplify: x - 5 = ±3i (Since the square root of -9 is 3i)

4. Solve for x: x = 5 ± 3i

The Solutions

Therefore, the solutions to the equation (x - 5)² = -9 are:

• x = 5 + 3i
• x = 5 - 3i

These are complex number solutions, representing points on the complex plane.