Difference between revisions of "1950 AHSME Problems/Problem 24"
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the two roots are <math>\sqrt{x-2}=-2, \sqrt{x-2}=1</math> | the two roots are <math>\sqrt{x-2}=-2, \sqrt{x-2}=1</math> | ||
− | notice, that the first root is extraneous as the range for a square root is always the non-negative numbers; thus, the only real root for <math>x</math> occurs for the second root; squaring both sides and solving for <math>x</math> gives <math> | + | notice, that the first root is extraneous as the range for a square root is always the non-negative numbers; thus, the only real root for <math>x</math> occurs for the second root; squaring both sides and solving for <math>x</math> gives <math>x=3 \Rightarrow \boxed{\textbf{(E)} \text{1 real root}}</math>$ |
== See Also == | == See Also == |
Revision as of 00:11, 10 June 2020
Contents
[hide]Problem
The equation has:
Solution 1
Original Equation
Subtract x from both sides
Square both sides
Get all terms on one side
Factor
If you put down A as your answer, it's wrong. You need to check for extraneous roots.
There is
Solution 2
It's not hard to note that simply works, as . But, is increasing, and is increasing, so is the only root. If , , and similarly if , then . Imaginary roots don't have to be considered because squaring the equation gives a quadratic, and quadratics either have only imaginary or only real roots.
Solution 3
We can create symmetry in the equation: let , then we have the two roots are
notice, that the first root is extraneous as the range for a square root is always the non-negative numbers; thus, the only real root for occurs for the second root; squaring both sides and solving for gives $
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
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