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

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== Problem==
 
== Problem==
  
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The equation <math>x + sqrt(x-2) = 4</math> has:
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The equation <math>x + \sqrt{x-2} = 4</math> has:
  
 
<math> \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} </math>
 
<math> \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} </math>
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 +
==Solution==
 +
<math>x + \sqrt{x-2} = 4</math> Original Equation
 +
 +
<math>\sqrt{x-2} = 4 - x</math> Subtract x from both sides
 +
 +
<math>x-2 = 16 - 8x + x^2</math> Square both sides
 +
 +
<math>x^2 - 9x + 18 = 0</math> Get all terms on one side
 +
 +
<math>(x-6)(x-3) = 0</math> Factor
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<math>x = \{6, 3\}</math>
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 +
If you put down A as your answer, it's wrong. You need to check for extraneous roots.
 +
 +
<math>6 + \sqrt{6 - 2} = 6 + \sqrt{4} = 6 + 2 = 8 \ne 4</math>
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<math>3 + \sqrt{3-2} = 3 + \sqrt{1} = 3 + 1 = 4 \checkmark</math>
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There is <math>\textbf{(E)} \text{1 real root}</math>
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{{AHSME box|year=1950|num-b=23|num-a=25}}

Revision as of 12:07, 29 December 2011

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

1950 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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All AHSME Problems and Solutions
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