Solving x2−⌊x⌋−4=0

Question

Solving x2−⌊x⌋−4=0

Korbin Ochoa · Accepted Answer

Step 1If&amp;nbsp;|x|≥3, then&amp;nbsp;x2&amp;gt;x+4≥⌊x⌋+4, and no solution is possible. Therefore the only solutions have⌊x⌋=−3,−2,−1,0,1,and 2. Let&#039;s see if&amp;nbsp;⌊x⌋=−3&amp;nbsp;works:x2−(−3)−4=0x=±1But neither&amp;nbsp;⌊1⌋&amp;nbsp;nor&amp;nbsp;⌊−1⌋&amp;nbsp;equals -3 so we have no solution hereNow try&amp;nbsp;⌊x⌋=−2x2−(−2)−4=0x=±2We see that&amp;nbsp;⌊2⌋≠−2&amp;nbsp;so that&#039;s not a solution, but&amp;nbsp;⌊−2⌋=−2&amp;nbsp;so we have a solution.Check the other 4 cases similarly and find that&amp;nbsp;x=6&amp;nbsp;is the other solution.

Esteban Sloan · Accepted Answer

Step 1You have shown the graph of the quadratic equation in your question. However, you have the cubic equation.x3=⌊x⌋+4Let, x&amp;gt;0 then⌊x⌋≤x&amp;lt;⌊x⌋+1(⌊x⌋)3−4≤x3−4&amp;lt;(⌊x⌋+1)3−4(⌊x⌋)3−4≤⌊x⌋&amp;lt;(⌊x⌋+1)3−4Let, ⌊x⌋=m&amp;gt;0,{(m+1)3−m−4&amp;gt;0m3−m−4≤0⟹{m3+3m2+2m−3&amp;gt;0m3−m−4≤0⇒m=1x3=5⇒x=35.I leave the rest of the x&amp;lt;0 case to you.

Solving \(\displaystyle{x}^{{2}}-\lfloor{x}\rfloor-{4}={0}\)

Answered question

Answer & Explanation

New Questions in Algebra I