Difference between revisions of "1950 AHSME Problems/Problem 24"

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There is <math>\textbf{(E)} \text{1 real root}</math>
 
There is <math>\textbf{(E)} \text{1 real root}</math>
  
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== See Also ==
 
{{AHSME box|year=1950|num-b=23|num-a=25}}
 
{{AHSME box|year=1950|num-b=23|num-a=25}}
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[[Category:Introductory Algebra Problems]]

Revision as of 14:20, 17 April 2012

Problem

The equation $x + \sqrt{x-2} = 4$ has:

$\textbf{(A)}\ 2\text{ real roots }\qquad\textbf{(B)}\ 1\text{ real and}\ 1\text{ imaginary root}\qquad\textbf{(C)}\ 2\text{ imaginary roots}\qquad\textbf{(D)}\ \text{ no roots}\qquad\textbf{(E)}\ 1\text{ real root}$

Solution

$x + \sqrt{x-2} = 4$ Original Equation

$\sqrt{x-2} = 4 - x$ Subtract x from both sides

$x-2 = 16 - 8x + x^2$ Square both sides

$x^2 - 9x + 18 = 0$ Get all terms on one side

$(x-6)(x-3) = 0$ Factor

$x = \{6, 3\}$

If you put down A as your answer, it's wrong. You need to check for extraneous roots.

$6 + \sqrt{6 - 2} = 6 + \sqrt{4} = 6 + 2 = 8 \ne 4$

$3 + \sqrt{3-2} = 3 + \sqrt{1} = 3 + 1 = 4 \checkmark$

There is $\textbf{(E)} \text{1 real root}$

See Also

1950 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AHSME Problems and Solutions