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The graphs of \(y=x^4\) and \(y=7x^2-10\) intersect at four points with \(x\)-coordinates \(\pm \sqrt{m}\)  and \(\pm \sqrt{n}\), where \(m > n\). What is \(m-n\) ?

mathtoo  Aug 22, 2018
edited by mathtoo  Aug 22, 2018
edited by mathtoo  Aug 22, 2018

Best Answer 

 #1
avatar+26965 
+4

Equate the two expressions:   x4 = 7x2 - 10 

 

Rearrange:   x4 - 7x2 + 10 = 0

 

Let z = x2 

 

z2 - 7z + 10 = 0

 

This factorises nicely as:  (z - 2)(z - 5) = 0

 

So z = 2 and z = 5

 

or  x2 = 2  and x2 = 5

 

n = 21/2, m = 51/2   so m - n = 51/2 - 21/2 

.

Alan  Aug 22, 2018
 #1
avatar+26965 
+4
Best Answer

Equate the two expressions:   x4 = 7x2 - 10 

 

Rearrange:   x4 - 7x2 + 10 = 0

 

Let z = x2 

 

z2 - 7z + 10 = 0

 

This factorises nicely as:  (z - 2)(z - 5) = 0

 

So z = 2 and z = 5

 

or  x2 = 2  and x2 = 5

 

n = 21/2, m = 51/2   so m - n = 51/2 - 21/2 

.

Alan  Aug 22, 2018
 #2
avatar+599 
+2

Thanks! So, the answer is 5-2=3?

mathtoo  Aug 22, 2018
 #3
avatar+26965 
+2

Yes.  I'm afraid I gave the positive roots of the equation the labels m and n, but, had I looked more carefully, I would have noticed that your definition of m and n referred to the values under the square root sign!

Alan  Aug 22, 2018
edited by Alan  Aug 22, 2018

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