Difference between revisions of "2007 iTest Problems/Problem 55"
Rockmanex3 (talk | contribs) (Solution to Problem 55 — typing this on mobile took a long time) |
Rockmanex3 (talk | contribs) m (Correction) |
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51525 &\bigcirc 51529 \\ | 51525 &\bigcirc 51529 \\ | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | This means that <math>15\sqrt{229}</math> is less than <math>227,</math> so <math>225 + 15\sqrt{229}</math> is less than <math>226</math>. Thus, the value of the greatest integer less than or equal to <math>x</math> is <math>\boxed{225}</math>. | + | This means that <math>15\sqrt{229}</math> is less than <math>227,</math> so <math>\tfrac{225 + 15\sqrt{229}}{2}</math> is less than <math>226</math>. Thus, the value of the greatest integer less than or equal to <math>x</math> is <math>\boxed{225}</math>. |
==See Also== | ==See Also== |
Revision as of 18:07, 3 August 2018
The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Problem
Let . Let be the smallest real solution of . Find the value of .
Solution
Plugging in results in We can solve the equation a bit easier by letting . Using the Quadratic Formula yields . However, we have to check for extraneous solutions and calculate the highest integer less than or equal to . This can be easily done using a calculator, but in this solution, we will not use a calculator.
We know that and , so . That means . Inserting the value into the equation would result in the right hand side having a square root of a negative number, so is an extraneous solution.
On the other hand, since we know that there is a solution to the equation and makes both sides positive in the original equation, is a valid solution. We know that is greater than , but we need to know if it’s less than
This means that is less than so is less than . Thus, the value of the greatest integer less than or equal to is .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 54 |
Followed by: Problem 56 | |
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