Common Misconceptions (Algebra)
What are the solution(s) to the equation \[\sqrt{x+ 4} = x -2?\]
(Partial solution)
The equation can be rewritten as follows:
\[\begin{align}
\sqrt{x+ 4} &= x -2\\
x+ 4 &= (x -2)^2\\
x+ 4 &= x^2 - 4x + 4\\
0 &= x^2 - 5x\\
0 &= x(x - 5).
\end{align}\]
Where is the error in the solution to this problem?
Find the solution set of \(20 - x > 5\).
Line 1. \(\,\,\,\,\,20 - x > 5\)
Line 2. \(\,\,\,\,\,-x > -15\)
Line 3. \(\,\,\,\,\,x >15\)
Where is the error in the solution to this problem?
Find the solution set of \(3 - x < 8\).
Step 1. \(\,\,\,\,\,3 - x < 8\)
Step 2. \(\,\,\,\,\,-x < 5\)
Step 3. \(\,\,\,\,\,x < -5\)
Is the following solution valid?
\(\,\,\,\,\,\sqrt{x - 5} = -3\)
\(\,\,\,\,\,x - 5 = 9\)
\(\,\,\,\,\,x = 14\)
Consider the following work done to solve the equation \[\sqrt{x + 1} = 1.\]
\(\,\,\,\,\,\sqrt{x} + \sqrt{1} = 1\)
\(\,\,\,\,\,\sqrt{x} + 1 = 1\)
\(\,\,\,\,\,\sqrt{x} = 0\)
\(\,\,\,\,\,x = 0\)