%5E2%3D-9.jpeg "(x+3)^2=-9")
Solving the Equation (x + 3)^2 = -9
This equation presents a unique challenge as it involves a square equaling a negative number. Let's break down how to solve it:
Understanding the Problem
- Squares are always non-negative: The square of any real number (positive, negative, or zero) is always greater than or equal to zero. This means there's no real number that, when squared, results in a negative value.
- Imaginary Numbers: To address this, we need to introduce the concept of imaginary numbers. The imaginary unit, denoted by 'i', is defined as the square root of -1 (i.e., i² = -1).
Solving for x
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Isolate the squared term: The equation (x + 3)² = -9 already has the squared term isolated.
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Take the square root of both sides: √((x + 3)²) = ±√(-9)
Remember: When taking the square root of both sides of an equation, we need to consider both positive and negative solutions. -
Simplify: x + 3 = ±√(-9)
x + 3 = ±3i (where 'i' is the imaginary unit) -
Solve for x: x = -3 ± 3i
Solutions
Therefore, the solutions to the equation (x + 3)² = -9 are:
- x = -3 + 3i
- x = -3 - 3i
These solutions are complex numbers, consisting of a real part (-3) and an imaginary part (±3i).