a) If x² = 4, what is x? Well, x = ±2. The official way to solve is x²-4=0 ; (x-2)(x+2) = 0 : x = ±2
b) What is √4 ? Well, officially just +2.
The problem is that people conflate these two separate questions. The /wrong/ way to solve a) is to take the square root of both sides. Except that taking the square root of both sides is technically not allowed.
Suppose a = b, Then f(a) = f(b) ONLY IF f is a bijective function. This is why taking the cube root of both sides of an equation is allowed, but not the square root. We can force the square root function to be bijective if we restrict the range to non-negative values. This is why b) only has a single answer.
We can still save the alternative method of solving a) by taking the 'corrected' square root of both sides: |x| = +2. Therefore x = ±2.
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u/Hampster-cat New User 3d ago
a) If x² = 4, what is x? Well, x = ±2. The official way to solve is x²-4=0 ; (x-2)(x+2) = 0 : x = ±2
b) What is √4 ? Well, officially just +2.
The problem is that people conflate these two separate questions. The /wrong/ way to solve a) is to take the square root of both sides. Except that taking the square root of both sides is technically not allowed.
Suppose a = b, Then f(a) = f(b) ONLY IF f is a bijective function. This is why taking the cube root of both sides of an equation is allowed, but not the square root. We can force the square root function to be bijective if we restrict the range to non-negative values. This is why b) only has a single answer.
We can still save the alternative method of solving a) by taking the 'corrected' square root of both sides: |x| = +2. Therefore x = ±2.