1960 AHSME Problems/Problem 39: Difference between revisions
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If <math>b</math> is not real, where <math>b = m+ni</math> and <math>n \neq 0</math>, then <math>\sqrt{-3b^2}</math> evaluates to <math>\sqrt{-3m^2 - 6mni + 3n^2}</math>. As long as <math>m \neq 0</math>, the expression can also be imaginary because a real number squared will be a real number. | If <math>b</math> is not real, where <math>b = m+ni</math> and <math>n \neq 0</math>, then <math>\sqrt{-3b^2}</math> evaluates to <math>\sqrt{-3m^2 - 6mni + 3n^2}</math>. As long as <math>m \neq 0</math>, the expression can also be imaginary because a real number squared will be a real number. | ||
From these two points, the answer is <math>\boxed{\textbf{(E)}}</math>. | From these two points, the answer is <math>\boxed{\textbf{(E)}}</math>. | ||
==Video Solution== | |||
https://youtu.be/ZdM2ou5Gsuw?t=312 | |||
~MathProblemSolvingSkills.com | |||
==See Also== | ==See Also== | ||
Latest revision as of 21:28, 28 December 2023
Problem
To satisfy the equation 
, 
and 
must be:
Solution
First, note that 
and 
. Cross multiply both sides to get
![\[a^2 + 2ab + b^2 = ab\]](http://latex.artofproblemsolving.com/c/e/0/ce049f4a6fe4c632a33226e51f59b9a54048ed9d.png)
Subtract both sides by 
to get
![\[a^2 + ab + b^2 = 0\]](http://latex.artofproblemsolving.com/9/f/e/9fef7cf665bb72b161fa423c6703eaf94bddf371.png)
From the quadratic formula,
![\[a = \frac{-b \pm \sqrt{b^2 - 4b^2}}{2}\]](http://latex.artofproblemsolving.com/5/1/0/510ea26a7484c93622f2e908c8de305aeea5a860.png)
![\[a = \frac{-b \pm \sqrt{-3b^2}}{2}\]](http://latex.artofproblemsolving.com/c/a/4/ca43c798dc91274a339a6be3a35254d3ce937564.png)
If is real, then 
is imaginary because 
is negative.
If is not real, where 
and 
, then evaluates to 
. As long as 
, the expression can also be imaginary because a real number squared will be a real number.
From these two points, the answer is 
.
Video Solution
https://youtu.be/ZdM2ou5Gsuw?t=312
~MathProblemSolvingSkills.com
See Also
| 1960 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 38 |
Followed by Problem 40 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
