Difference between revisions of "2014 AMC 10B Problems/Problem 20"
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==Solution 1== | ==Solution 1== | ||
− | First, note that <math>50+1=51</math>, which motivates us to factor the polynomial as <math>(x^2-50)(x^2-1)</math>. Since this expression is negative, one term must be negative and the other positive. Also, the first term is obviously smaller than the second, so <math>x^2-50<0<x^2-1</math>. Solving this inequality, we find <math>1<x^2<50</math>. There are exactly <math>12</math> integers <math>x</math> that satisfy this inequality, <math>\pm 2,3,4,5,6,7</math>. | + | First, note that <math>50+1=51</math>, which motivates us to factor the polynomial as <math>(x^2-50)(x^2-1)</math>. Since this expression is negative, one term must be negative and the other positive. Also, the first term is obviously smaller than the second, so <math>x^2-50<0<x^2-1</math>. Solving this inequality, we find <math>1<x^2<50</math>. There are exactly <math>12</math> integers <math>x</math> that satisfy this inequality, <math>\pm \{2,3,4,5,6,7\}</math>. |
Thus our answer is <math>\boxed{\textbf {(C) } 12}</math>. | Thus our answer is <math>\boxed{\textbf {(C) } 12}</math>. | ||
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==Solution 3 (Graph)== | ==Solution 3 (Graph)== | ||
As with Solution <math>1</math>, note that the quartic factors to <math>(x^2-50)\cdot(x^2-1)</math>, which means that it has roots at <math>-5\sqrt{2}</math>, <math>-1</math>, <math>1</math>, and <math>5\sqrt{2}</math>. Now, because the original equation is of an even degree and has a positive leading coefficient, both ends of the graph point upwards, meaning that the graph dips below the <math>x</math>-axis between <math>-5\sqrt{2}</math> and <math>-1</math> as well as <math>1</math> and <math>5\sqrt{2}</math>. <math>5\sqrt{2}</math> is a bit more than <math>7</math> (<math>1.4\cdot 5=7</math>) and therefore means that <math> -7,-6,-5,-4,-3,-2,2,3,4,5,6,7</math> all give negative values. | As with Solution <math>1</math>, note that the quartic factors to <math>(x^2-50)\cdot(x^2-1)</math>, which means that it has roots at <math>-5\sqrt{2}</math>, <math>-1</math>, <math>1</math>, and <math>5\sqrt{2}</math>. Now, because the original equation is of an even degree and has a positive leading coefficient, both ends of the graph point upwards, meaning that the graph dips below the <math>x</math>-axis between <math>-5\sqrt{2}</math> and <math>-1</math> as well as <math>1</math> and <math>5\sqrt{2}</math>. <math>5\sqrt{2}</math> is a bit more than <math>7</math> (<math>1.4\cdot 5=7</math>) and therefore means that <math> -7,-6,-5,-4,-3,-2,2,3,4,5,6,7</math> all give negative values. | ||
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+ | ==Solution 4== | ||
+ | Let <math>x^{2}=u</math>. Then the expression becomes <math>u^{2}-51u+50</math> which can be factored as <math>\left(u-1\right)\left(u-50\right)</math>. Since the expression is negative, one of <math>\left(u-1\right)</math> and <math>\left(u-50\right)</math> need to be negative. <math>u-1>u-50</math>, so <math>u-1>0</math> and <math>u-50<0</math>, which yields the inequality <math>50>u>1</math>. Remember, since <math>u=x^{2}</math> where <math>x</math> is an integer, this means that <math>u</math> is a perfect square between <math>1</math> and <math>50</math>. There are <math>6</math> values of <math>u</math> that satisfy this constraint, namely <math>4</math>, <math>9</math>, <math>16</math>, <math>25</math>, <math>36</math>, and <math>49</math>. Solving each of these values for <math>x</math> yields <math>12</math> values (as <math>x</math> can be negative or positive) <math>\Longrightarrow \boxed{\textbf {(C) } 12}</math>. | ||
+ | ~JH. L | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2014|ab=B|num-b=19|num-a=21}} | {{AMC10 box|year=2014|ab=B|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:18, 18 June 2022
Problem
For how many integers is the number negative?
Solution 1
First, note that , which motivates us to factor the polynomial as . Since this expression is negative, one term must be negative and the other positive. Also, the first term is obviously smaller than the second, so . Solving this inequality, we find . There are exactly integers that satisfy this inequality, .
Thus our answer is .
Solution 2
Since the part of has to be less than (because we want to be negative), we have the inequality . has to be positive, so is negative. Then we have . We know that if we find a positive number that works, it's parallel negative will work. Therefore, we just have to find how many positive numbers work, then multiply that by . If we try , we get , and therefore doesn't work. Test two on your own, and then proceed. Since two works, all numbers above that satisfy work, that is the set {}. That equates to numbers. Since each numbers' negative counterparts work, .
Solution 3 (Graph)
As with Solution , note that the quartic factors to , which means that it has roots at , , , and . Now, because the original equation is of an even degree and has a positive leading coefficient, both ends of the graph point upwards, meaning that the graph dips below the -axis between and as well as and . is a bit more than () and therefore means that all give negative values.
Solution 4
Let . Then the expression becomes which can be factored as . Since the expression is negative, one of and need to be negative. , so and , which yields the inequality . Remember, since where is an integer, this means that is a perfect square between and . There are values of that satisfy this constraint, namely , , , , , and . Solving each of these values for yields values (as can be negative or positive) . ~JH. L
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 | ||
All AMC 10 Problems and Solutions |
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