Difference between revisions of "1953 AHSME Problems/Problem 44"
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+ | 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>. | ||
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+ | 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>. | ||
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+ | ==See Also== | ||
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+ | {{AHSME 50p box|year=1953|num-b=43|num-a=45}} | ||
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+ | {{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 | |
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All AHSME Problems and Solutions |
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