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

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==Solution 2==
 
==Solution 2==
  
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It's not hard to note that x=3 simply works, as <math>3 + \sqrt{1} = 4</math>. But, <math>x</math> is increasing, and <math>\sqrt{x-2}</math> is increasing, so <math>3</math> is the only root. If <math>x</math> is smaller than <math>3</math>, <math>x + \sqrt{x-2} < 4</math>, and similarly if <math>x > 3</math>, then <math>x + \sqrt{x-2} > 4</math>. 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.
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It's not hard to note that <math>x=3</math> simply works, as <math>3 + \sqrt{1} = 4</math>. But, <math>x</math> is increasing, and <math>\sqrt{x-2}</math> is increasing, so <math>3</math> is the only root. If <math>x</math> is smaller than <math>3</math>, <math>x + \sqrt{x-2} < 4</math>, and similarly if <math>x > 3</math>, then <math>x + \sqrt{x-2} > 4</math>. 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.
  
 
== See Also ==
 
== See Also ==

Revision as of 16:04, 11 May 2019

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 1

$x + \sqrt{x-2} = 4$ 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 $\boxed{\textbf{(E)} \text{1 real root}}$

Solution 2

It's not hard to note that $x=3$ simply works, as $3 + \sqrt{1} = 4$. But, $x$ is increasing, and $\sqrt{x-2}$ is increasing, so $3$ is the only root. If $x$ is smaller than $3$, $x + \sqrt{x-2} < 4$, and similarly if $x > 3$, then $x + \sqrt{x-2} > 4$. 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.

See Also

1950 AHSC (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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