Difference between revisions of "1980 AHSME Problems/Problem 8"
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<math>\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0</math> | <math>\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0</math> | ||
<math>\text{(E)} \ \text{two pairs for each} ~b \neq 0</math> | <math>\text{(E)} \ \text{two pairs for each} ~b \neq 0</math> | ||
+ | |||
+ | == Solution == | ||
+ | |||
+ | Adding the two fractions on the left side yields <math>\frac{a+b}{ab}=\frac{1}{a+b}</math>. | ||
+ | |||
+ | Cross-multiplying yields <math>a^2+2ab+b^2=ab</math>. | ||
+ | |||
+ | Subtracting <math>ab</math> from both sides yields <math>a^2+ab+b^2=0</math>. | ||
+ | |||
+ | Suppose <math>b</math> is constant and <math>a</math> is variable. Then the discriminant of this quadratic equation is <math>b^2 - 4 \cdot 1 \cdot b^2 = -3b^2</math>, which is negative if <math>b \neq 0</math>. Therefore, for each <math>b \neq 0</math>, this equation has no real solutions, and the answer is <math>\boxed{\text{(\textbf{A})\ none}}</math>. | ||
+ | |||
+ | == See also == | ||
+ | {{AHSME box|year=1980|num-b=7|num-a=9}} | ||
+ | {{MAA Notice}} |
Latest revision as of 02:36, 23 July 2025
Problem
How many pairs of non-zero real numbers satisfy the equation
Solution
Adding the two fractions on the left side yields .
Cross-multiplying yields .
Subtracting from both sides yields
.
Suppose is constant and
is variable. Then the discriminant of this quadratic equation is
, which is negative if
. Therefore, for each
, this equation has no real solutions, and the answer is
.
See also
1980 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 | ||
All AHSME Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.