Difference between revisions of "1953 AHSME Problems/Problem 44"
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==Solution== | ==Solution== | ||
+ | |||
+ | Let <math>x^2+bx+c=0</math> represent the correct equation. Since the coefficient of the <math>x^2</math> term is <math>1</math>, the sum of the roots is <math>-b</math>, and the product of the roots is <math>c</math>. | ||
+ | |||
+ | If a student only misreads the constant term, he must have the correct sum of roots. Therefore, the sum of the roots is <math>8+2=10</math>, so <math>b=-10</math>. If a student only misreads the linear term, he must have the correct product of the roots. The product of the roots is <math>(-9)\cdot (-1) = 9</math>, so <math>c=9</math>. The correct equation is <math>\boxed{\textbf{(A) } x^2-10x+9=0}</math>. | ||
==See Also== | ==See Also== | ||
− | {{AHSME 50p box|year=1953|num-b= | + | {{AHSME 50p box|year=1953|num-b=43|num-a=45}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:56, 14 February 2020
Problem
In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and
obtains and
for the roots. Another student makes a mistake only in the coefficient of the first degree term and
find
and
for the roots. The correct equation was:
Solution
Let represent the correct equation. Since the coefficient of the
term is
, the sum of the roots is
, and the product of the roots is
.
If a student only misreads the constant term, he must have the correct sum of roots. Therefore, the sum of the roots is , so
. If a student only misreads the linear term, he must have the correct product of the roots. The product of the roots is
, so
. The correct equation is
.
See Also
1953 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 43 |
Followed by Problem 45 | |
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 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
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