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Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{2}$ has exactly one real solution.

 Aug 7, 2023
 #1
avatar+161 
+1

Note: This sounds suspiciously like an Alcumus problem that I've done before.

 

This is just a quadratic. \(\frac{1}{2x} = \frac{r-x}{2}\)

So, \(2 = 2xr-2x^2\)

\(2x^2-2xr+2 = 0\)

Now just plug in the quadratic formula and you should get the answer, don't forget to multiply.

 

P.S I left some of the problem for you to do, just to make sure that you are not cheating

 

Hope this helped, EJ

 

Why my posts always flag for moderation Cphil?

 Aug 7, 2023
edited by Imcool  Aug 7, 2023
 #3
avatar+125627 
+1

I don't know why!!!

 

I just returned  to this site in the last few days and some strange things are  going  on.....anyway....I "unblocked" your answer !!!!

 

 

cool cool cool

CPhill  Aug 7, 2023
edited by CPhill  Aug 7, 2023
 #4
avatar+161 
0

Thanks Cphil for the unblock, just posted a few more answers, all of them are blocked. Oof lol

Imcool  Aug 7, 2023
 #5
avatar+36654 
0

Yah....mine are all blocked too......

~EP

ElectricPavlov  Aug 7, 2023
 #6
avatar+125627 
0

OK, EP

 

I'll  "unblock" yours, as  well !!!

 

This is  crazy  !!!

 

cool cool cool

CPhill  Aug 7, 2023
 #2
avatar+125627 
+1

1/ (2x)  = ( r - x) / 2         cross-multiply

 

2 = 2x ( r - x)

 

2 = 2xr -2x^2       rearrange as

 

2x^2 -2xr + 2 = 0    simplify

 

x^2 - xr + 1  =   0

 

This will  have  one real solution  when

 

r^2 - 4(1)(1) =  0

 

r^2  - 4  = 0

 

r^2 = 4             take both  roots

 

r = 2 , -2

 

And the  product of these =  -4

 

 

cool cool cool

 Aug 7, 2023
 #7
avatar+2 
0

We can solve this problem using the discriminant of the quadratic equation. For a quadratic equation of the form ax^2 + bx + c = 0, the discriminant is b^2 - 4ac. If the discriminant is positive, the quadratic equation has two distinct real roots. If the discriminant is zero, the quadratic equation has one real root. If the discriminant is negative, the quadratic equation has no real roots.

In this problem, we are given the quadratic equation x^2 - (a + 1)x + a = 0, and we want to find the product of all values of a for which the equation has exactly one real solution.

Using the quadratic formula, we can solve for x:

x = [(a + 1) ± sqrt((a + 1)^2 - 4a)] / 2

The equation has exactly one real solution when the discriminant is zero:

(a + 1)^2 - 4a = 0

Simplifying this equation, we get:

a^2 - 2a + 1 = 0

This equation has a double root at a = 1. Therefore, the product of all values of a for which the equation has exactly one real solution is 1.

 Aug 7, 2023

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