@Pranjal Jain
–
One can easily draw the grpah of \(\frac{x^2}{10} + \frac{51}{40}\) and \(\left \lfloor x \right \rfloor\) and find the intersection.
–
Kartik Sharma
·
1 year ago

@Dev Sharma
–
for ease write the equation as\[(2x)^2=40[x]-51\] by definition the floor function is an integer, implying 40[x]-51 is also an integer which also implies \((2x)^2\) is an integer
–
Aareyan Manzoor
·
1 year ago

All the real roots of the given equation will lie in (1,9). This can be easily figured out once you draw the graphs: \(y=4x^2+51\) and \(y=40\lfloor x \rfloor\) [because the points at which these two graphs will intersect will be the solutions of the given equation] and observe that the later one will never be able to catch the former one.

So now let us look at our equation, which can be rewritten as \(4x^2+51=40 \lfloor x \rfloor\). Looking at the RHS we conclude that it is always an integer. Therefore for the equation to be correct LHS must be an integer too.

Now put \(\lfloor x \rfloor = i\) in the RHS, where \(i\) is an integer lying in \((1,9)\) and find the value of \(x\). To be more explicit, only those values of \(x\) will be considered as the roots of the given equation which will satisfy \(\lfloor x \rfloor = i\) simultaneously.
–
Sandeep Bhardwaj
·
1 year ago

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Actually, the solution is simpler than you might think.

Rearranging the equation:
\[40\lfloor x\rfloor=4x^2+51 \]
\[40(x-1)<4x^2+51\leq40x\]

From the lower limit,
\[4x^2+51>40(x-1) \]
\[(x-\frac{7}{2})(x-\frac{13}{2})>0 \]

(\(x<\frac{7}{2}\) and \(x< \frac{13}{2} \))
OR
(\( x>\frac{7}{2}\) and \( x>\frac{13}{2}\) )
which imply \( x<\frac{7}{2} \) OR \( x>\frac{13}{2} \)

Similarly, from upper limit,
\[4x^2+51\leq 40x \]
\[(x-\frac{17}{2})(x-\frac{3}{2})\leq 0 \]

(\(x\geq \frac{17}{2}\) and \(x \leq \frac{3}{2} \))
OR
(\( x\leq \frac{17}{2}\) and \( x\geq \frac{3}{2}\) )
which imply (an impossible case) OR \( \frac{3}{2} \leq x \leq \frac{17}{2} \)

Hence, solutions are in the region
\[(1.5\leq x<3.5)U( 6.5<x\leq8.5) \]

Rearranging the given equation, we get
\[x=\frac{\sqrt{40\lfloor x \rfloor-51}}{2}\]

Solving in each region,

\[@ 1.5\leq x<2, \lfloor x \rfloor=1\]
x has no real solutions

\[@ 2\leq x<3, \lfloor x \rfloor=2\]
\[x=\frac{\sqrt{29}}{2}\]

\[@ 3\leq x<3.5, \lfloor x \rfloor=3\]
\[x=\frac{\sqrt{69}}{2}\]
Which is a contradiction (as it is not in the region)

(Note: I may even have done some stupid mistake, please verify)

\[@ 6.5<x<7, \lfloor x \rfloor=6\]
\[x=\frac{3}{2}\sqrt{21}\]

\[@ 7\leq x<8, \lfloor x \rfloor=7\]
\[x=\frac{\sqrt{229}}{2}\]

\[@ 8\leq x<8.5, \lfloor x \rfloor=8\]
\[x=\frac{\sqrt{269}}{2}\]

the solutions are
\[x=\frac{\sqrt{29}}{2},\frac{3}{2}\sqrt{21},\frac{\sqrt{229}}{2},\frac{\sqrt{269}}{2}\]

[Please comment, if you find any of the assumptions to be wrong or questionable. ]
–
Harish Sasikumar
·
1 year ago

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@Harish Sasikumar
–
Can someone explain, why there are 2 'rejectable' solutions coming up as we are solving it ?
–
Harish Sasikumar
·
1 year ago

@Silver Vice
–
How did you get that solution? It doesn't seem correct.
\[ 17/2=8.5\]

\[ LHS = 4(8.5)^2 -40\lfloor 8.5 \rfloor +51 =289-320+51=20\neq 0\]
It doesn't satisfy the equation
–
Harish Sasikumar
·
1 year ago

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@Harish Sasikumar
–
I used the Shreedhacharya rule. it gave me two roots, one is 3/2 as mentioned in your answer. the other was 17/2. I am an amateur at math and might have made some mistake. Joined brilliant.org recently to improve my problem solving skill.
–
Silver Vice
·
1 year ago

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@Silver Vice
–
OK. Understood the mistake. You took it as a quadratic equation and solved. That's how you got those solutions (3/2 and 17/2).

However it is NOT a quadratic equation. There is a 'floor' function in between which is represented as \(\lfloor x \rfloor \) and is defined as the the integral part of a real number. For example
\[\lfloor 3.4 \rfloor =3\]

Keep doing problems Brilliant. It's an awesome source of problems. Best of luck :).
–
Harish Sasikumar
·
1 year ago

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@Harish Sasikumar
–
Floor function? ill research about it. Ill surely try to encounter more such questions. Thanks ! :)
–
Silver Vice
·
1 year ago

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How about 4 \(y\) - 40 \(\lfloor \sqrt{y}\rfloor\) + 51 = 0 for all rational y?

y = {7.25, 47.25, 57.25, 67.25}

\(z\) - 40 \(\lfloor \sqrt{\frac{z}{4}}\rfloor \) + 51 = 0 for all positive integer z!

z = {29, 189, 229, 269}

I guess this is the solution of the equation 4 \(x^2\) - 40 \(\lfloor x \rfloor\) + 51 = 0.

This means it doesn't follow only 2 but duplicated into 4. Changed from {\(\frac{17}{2}, \frac32\)}.

\(z\) - 40 \(\lfloor -\sqrt{\frac{z}{4}}\rfloor \) + 51 = 0 for all positive integer z could be some more but not found with any.

x = {\(\frac{\sqrt{29}}{2}, \frac{3 \sqrt{21}}{2}, \frac{\sqrt{229}}{2}, \frac{\sqrt{269}}{2}\)}
–
Lu Chee Ket
·
1 year ago

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obviously [x]≥2 as \(2x=\sqrt{40[x]-51}\) and x real.
put x =[x]+y where y is frac part ≥0 and <1.so
\[4([x]+y)^2-40[x]+51=4y^2+8[x]y+(4[x]^2-40[x]+51)=0\]
due to y range:(the other one will always <0 on the restrictions)
\[0≤\dfrac{-2[x]+\sqrt{40[x]-51}}{2}<1\]
\[0≤-2[x]+\sqrt{40[x]-51}<2\]
\[2[x]≤\sqrt{40[x]-51}<2+2[x]\]
\[4[x]^2≤40[x]-51<4[x]^2+8[x]+4\]
\[4[x]^2-40[x]+51≤0<4[x]^2-32[x]+55\]
different cases we get(since all int≥2):
\[2≤[x]≤8 \cap [x]≤2(\text{squeezes with [x]≥2 to get [x]=2}) \quad or \quad [x]≥6\]
so \[\begin{cases}
[x]=2\\
6≤[x]≤8\end{cases}\]
just check all 4 cases to get respective y and x.
–
Aareyan Manzoor
·
1 year ago

## Comments

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TopNewest\((2x)^2- 40 \lfloor x \rfloor +51=0\)

Two things to be noticed here:

Therefore, the fractional part of \(x\) would be 0.5. Now we can write \(x=N+0.5\) and solve it quite easily.

Edit:@Dev Sharma @Aareyan Manzoor @Adarsh Kumar @Akshat Sharda This solution is wrong. Because \(2x\) need not be an integer. \((2x)^2\in Z\not \Rightarrow 2x\in Z\). I'll have to think again about an alternative method. – Pranjal Jain · 1 year agoLog in to reply

– Kartik Sharma · 1 year ago

One can easily draw the grpah of \(\frac{x^2}{10} + \frac{51}{40}\) and \(\left \lfloor x \right \rfloor\) and find the intersection.Log in to reply

– Dev Sharma · 1 year ago

why 2x must be integer?Log in to reply

– Aareyan Manzoor · 1 year ago

for ease write the equation as\[(2x)^2=40[x]-51\] by definition the floor function is an integer, implying 40[x]-51 is also an integer which also implies \((2x)^2\) is an integerLog in to reply

– Dev Sharma · 1 year ago

thanksLog in to reply

– Adarsh Kumar · 1 year ago

Ooops!I didn't see that !Thanx a lot!Log in to reply

A simple standard approach will crush it:

All the real roots of the given equation will lie in (1,9). This can be easily figured out once you draw the graphs: \(y=4x^2+51\) and \(y=40\lfloor x \rfloor\) [because the points at which these two graphs will intersect will be the solutions of the given equation] and observe that the later one will never be able to catch the former one.

So now let us look at our equation, which can be rewritten as \(4x^2+51=40 \lfloor x \rfloor\). Looking at the RHS we conclude that it is always an integer. Therefore for the equation to be correct LHS must be an integer too.

Now put \(\lfloor x \rfloor = i\) in the RHS, where \(i\) is an integer lying in \((1,9)\) and find the value of \(x\). To be more explicit, only those values of \(x\) will be considered as the roots of the given equation which will satisfy \(\lfloor x \rfloor = i\) simultaneously. – Sandeep Bhardwaj · 1 year ago

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Actually, the solution is simpler than you might think.

Rearranging the equation: \[40\lfloor x\rfloor=4x^2+51 \]

\[40(x-1)<4x^2+51\leq40x\]

From the lower limit, \[4x^2+51>40(x-1) \] \[(x-\frac{7}{2})(x-\frac{13}{2})>0 \]

(\(x<\frac{7}{2}\) and \(x< \frac{13}{2} \)) OR (\( x>\frac{7}{2}\) and \( x>\frac{13}{2}\) ) which imply \( x<\frac{7}{2} \) OR \( x>\frac{13}{2} \)

Similarly, from upper limit, \[4x^2+51\leq 40x \] \[(x-\frac{17}{2})(x-\frac{3}{2})\leq 0 \]

(\(x\geq \frac{17}{2}\) and \(x \leq \frac{3}{2} \)) OR (\( x\leq \frac{17}{2}\) and \( x\geq \frac{3}{2}\) ) which imply (an impossible case) OR \( \frac{3}{2} \leq x \leq \frac{17}{2} \)

Hence, solutions are in the region \[(1.5\leq x<3.5)U( 6.5<x\leq8.5) \]

Rearranging the given equation, we get \[x=\frac{\sqrt{40\lfloor x \rfloor-51}}{2}\]

Solving in each region,

\[@ 1.5\leq x<2, \lfloor x \rfloor=1\] x has no real solutions

\[@ 2\leq x<3, \lfloor x \rfloor=2\] \[x=\frac{\sqrt{29}}{2}\]

\[@ 3\leq x<3.5, \lfloor x \rfloor=3\] \[x=\frac{\sqrt{69}}{2}\] Which is a contradiction (as it is not in the region)

(Note: I may even have done some stupid mistake, please verify)

\[@ 6.5<x<7, \lfloor x \rfloor=6\] \[x=\frac{3}{2}\sqrt{21}\]

\[@ 7\leq x<8, \lfloor x \rfloor=7\] \[x=\frac{\sqrt{229}}{2}\]

\[@ 8\leq x<8.5, \lfloor x \rfloor=8\] \[x=\frac{\sqrt{269}}{2}\]

the solutions are\[x=\frac{\sqrt{29}}{2},\frac{3}{2}\sqrt{21},\frac{\sqrt{229}}{2},\frac{\sqrt{269}}{2}\][Please comment, if you find any of the assumptions to be wrong or questionable. ] – Harish Sasikumar · 1 year ago

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– Harish Sasikumar · 1 year ago

Can someone explain, why there are 2 'rejectable' solutions coming up as we are solving it ?Log in to reply

– Lu Chee Ket · 1 year ago

8.20060973342836Log in to reply

– Harish Sasikumar · 1 year ago

Yes, you are correct. I changed it.Log in to reply

– Lu Chee Ket · 1 year ago

Agree!Log in to reply

– Silver Vice · 1 year ago

I found 17/2 to be an answer too.Log in to reply

\[ LHS = 4(8.5)^2 -40\lfloor 8.5 \rfloor +51 =289-320+51=20\neq 0\] It doesn't satisfy the equation – Harish Sasikumar · 1 year ago

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– Silver Vice · 1 year ago

I used the Shreedhacharya rule. it gave me two roots, one is 3/2 as mentioned in your answer. the other was 17/2. I am an amateur at math and might have made some mistake. Joined brilliant.org recently to improve my problem solving skill.Log in to reply

However it is NOT a quadratic equation. There is a 'floor' function in between which is represented as \(\lfloor x \rfloor \) and is defined as the the integral part of a real number. For example \[\lfloor 3.4 \rfloor =3\]

Keep doing problems Brilliant. It's an awesome source of problems. Best of luck :). – Harish Sasikumar · 1 year ago

Log in to reply

– Silver Vice · 1 year ago

Floor function? ill research about it. Ill surely try to encounter more such questions. Thanks ! :)Log in to reply

How about 4 \(y\) - 40 \(\lfloor \sqrt{y}\rfloor\) + 51 = 0 for all rational y?

y = {7.25, 47.25, 57.25, 67.25}

\(z\) - 40 \(\lfloor \sqrt{\frac{z}{4}}\rfloor \) + 51 = 0 for all positive integer z!

z = {29, 189, 229, 269}

I guess this is the solution of the equation 4 \(x^2\) - 40 \(\lfloor x \rfloor\) + 51 = 0.

x = {\(\sqrt{\frac{29}{4}}, \sqrt{\frac{189}{4}}, \sqrt{\frac{229}{4}}, \sqrt{\frac{269}{4}}\)} = {\(\frac{\sqrt{29}}{2}, \frac{3 \sqrt{21}}{2}, \frac{\sqrt{229}}{2}, \frac{\sqrt{269}}{2}\)}

This means it doesn't follow only 2 but duplicated into 4. Changed from {\(\frac{17}{2}, \frac32\)}.

\(z\) - 40 \(\lfloor -\sqrt{\frac{z}{4}}\rfloor \) + 51 = 0 for all positive integer z could be some more but not found with any.

x = {\(\frac{\sqrt{29}}{2}, \frac{3 \sqrt{21}}{2}, \frac{\sqrt{229}}{2}, \frac{\sqrt{269}}{2}\)} – Lu Chee Ket · 1 year ago

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obviously [x]≥2 as \(2x=\sqrt{40[x]-51}\) and x real. put x =[x]+y where y is frac part ≥0 and <1.so \[4([x]+y)^2-40[x]+51=4y^2+8[x]y+(4[x]^2-40[x]+51)=0\] due to y range:(the other one will always <0 on the restrictions) \[0≤\dfrac{-2[x]+\sqrt{40[x]-51}}{2}<1\] \[0≤-2[x]+\sqrt{40[x]-51}<2\] \[2[x]≤\sqrt{40[x]-51}<2+2[x]\] \[4[x]^2≤40[x]-51<4[x]^2+8[x]+4\] \[4[x]^2-40[x]+51≤0<4[x]^2-32[x]+55\] different cases we get(since all int≥2): \[2≤[x]≤8 \cap [x]≤2(\text{squeezes with [x]≥2 to get [x]=2}) \quad or \quad [x]≥6\] so \[\begin{cases} [x]=2\\ 6≤[x]≤8\end{cases}\] just check all 4 cases to get respective y and x. – Aareyan Manzoor · 1 year ago

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