Difference between revisions of "2007 iTest Problems/Problem 36"
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Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. | Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation <math>x^4+25b^2=(4b^2-10b)x^2</math> has <math>\textit{at least}</math> two distinct real solutions. Find the value of <math>m+n</math>. | ||
| − | == Solution == | + | ==Solution== |
| + | |||
| + | The equation has quadratic form, so complete the square to solve for x. | ||
| + | |||
| + | <cmath>x^4 - (4b^2 - 10b)x^2 + 25b^2 = 0</cmath> | ||
| + | <cmath>x^4 - (4b^2 - 10b)x^2 + (2b^2 - 5b)^2 - 4b^4 + 20b^3 = 0</cmath> | ||
| + | <cmath>(x^2 - (2b^2 - 5b))^2 = 4b^4 - 20b^3</cmath> | ||
| + | |||
| + | In order for the equation to have real solutions, | ||
| + | |||
| + | <cmath>16b^4 - 80b^3 \ge 0</cmath> | ||
| + | <cmath>b^3(b - 5) \ge 0</cmath> | ||
| + | <cmath>b \le 0 \text{or } b \ge 5</cmath> | ||
| + | |||
| + | Note that <math>2b^2 - 5b = b(2b-5)</math> is greater than or equal to <math>0</math> when <math>b \le 0</math> or <math>b \ge 5</math>. Also, if <math>b = 0</math>, then expression leads to <math>x^4 = 0</math> and only has one unique solution, so discard <math>b = 0</math> as a solution. The rest of the values leads to <math>b^2</math> equalling some positive value, so these values will lead to two distinct real solutions. | ||
| + | |||
| + | Therefore, in interval notation, <math>b \in [-17,0) \cup [5,17]</math>, so the probability that the equation has at least two distinct real solutions when <math>b</math> is randomly picked from interval <math>[-17,17]</math> is <math>\frac{29}{34}</math>. This means that <math>m+n = \boxed{63}</math>. | ||
| + | |||
| + | ==See Also== | ||
| + | {{iTest box|year=2007|num-b=35|num-a=37}} | ||
| + | |||
| + | [[Category:Intermediate Algebra Problems]] | ||
| + | [[Category:Intermediate Probability Problems]] | ||
Revision as of 21:36, 14 June 2018
Problem
Let b be a real number randomly selected from the interval ![$[-17,17]$](http://latex.artofproblemsolving.com/2/e/5/2e548303b5cf904a5708fd8a4c7a575cde599a9b.png)
. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation 
has 
two distinct real solutions. Find the value of 
.
Solution
The equation has quadratic form, so complete the square to solve for x.
![\[x^4 - (4b^2 - 10b)x^2 + 25b^2 = 0\]](http://latex.artofproblemsolving.com/d/1/7/d1774d24f970655b699e26557ee8364f9ba73705.png)
![\[x^4 - (4b^2 - 10b)x^2 + (2b^2 - 5b)^2 - 4b^4 + 20b^3 = 0\]](http://latex.artofproblemsolving.com/f/6/a/f6a0c2bf6322f15e83e1d3f96b11e7f9d5c86a72.png)
![\[(x^2 - (2b^2 - 5b))^2 = 4b^4 - 20b^3\]](http://latex.artofproblemsolving.com/9/f/a/9faa07d441c20010aa93b001a2e84262e4c1e9fa.png)
In order for the equation to have real solutions,
![\[16b^4 - 80b^3 \ge 0\]](http://latex.artofproblemsolving.com/0/9/9/099efd5ee6eeb42593262c5527c7f1988089f84c.png)
![\[b^3(b - 5) \ge 0\]](http://latex.artofproblemsolving.com/0/6/a/06aaddb4ed0505705303df3791847188db2a8de4.png)
![\[b \le 0 \text{or } b \ge 5\]](http://latex.artofproblemsolving.com/d/b/d/dbd24e4345a66491132966e7924be6014bda0b52.png)
Note that 
is greater than or equal to 
when 
or 
. Also, if 
, then expression leads to 
and only has one unique solution, so discard as a solution. The rest of the values leads to 
equalling some positive value, so these values will lead to two distinct real solutions.
Therefore, in interval notation, ![$b \in [-17,0) \cup [5,17]$](http://latex.artofproblemsolving.com/0/1/9/01932bb7ac9e6791242020a7e33c948e4562b359.png)
, so the probability that the equation has at least two distinct real solutions when 
is randomly picked from interval is 
. This means that 
.
See Also
| 2007 iTest (Problems, Answer Key) | ||
| Preceded by: Problem 35 |
Followed by: Problem 37 | |
| 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 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4 | ||
