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Connecting Points

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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

#1
+26965
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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

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Alan  Aug 22, 2018
#1
+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
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Thanks! So, the answer is 5-2=3?

mathtoo  Aug 22, 2018
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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