if (a/b)=(c/d) then how is (a+b)/(c+d)=sqrt(ab/cd)

[inanisumeet]

I found a question in math magazine that said
if ratio of roots of ax²+2bx+c=0 is same as ratio of roots of px²+2qx+r=0 , then
(1)2b/ac=q²/pr
(2)b/ac=q/pr
(3)b²/ac=q²/pr
(4)None of these

Answer is given as (3) and the hint is

How did they make the last statement in above picture ?

Thanks.

 

[Dr.Peterson]

I found a question in math magazine that said
if ratio of roots of ax²+2bx+c=0 is same as ratio of roots of px²+2qx+r=0 , then
(1)2b/ac=q²/pr
(2)b/ac=q/pr
(3)b²/ac=q²/pr
(4)None of these

Answer is given as (3) and the hint is View attachment 25330

How did they make the last statement in above picture ?

Thanks.

Suppose that a = kc and b = kd. Expand (a+b)/(c+d) and sqrt((ab)/(cd)).

 

[Steven G]

a matches with p,
b matches with q and
c matches with r

Which choices are equals if you make the changes above. For example ac/b becomes pr/q (since I replaced a with p, c with r and b with q).

 

[JeffM]

Are the Greek letter the roots?

 

[Cubist]

How did they make the last statement in above picture ?

Substitute the following

[math]\gamma=\frac{\alpha\cdot \delta}{\beta}[/math]
into...

[math]\frac{\alpha+\beta}{\gamma+\delta}[/math]
and simplify.

Then use the fact that [math]\frac{z}{y} = \sqrt{\frac{z}{y}\times\frac{z}{y}}\,[/math] but only if [math]\frac{z}{y}\ge0[/math]

 

[inanisumeet]

Suppose that a = kc and b = kd. Expand (a+b)/(c+d) and sqrt((ab)/(cd)).

yes , in that case statement is valid.
If we say 2/3=(-2)/(-3) which is not equal to (2+(-2))/(3+(-3))

 

[inanisumeet]

Are the Greek letter the roots?

yes

 

[Dr.Peterson]

yes , in that case statement is valid.
If we say 2/3=(-2)/(-3) which is not equal to (2+(-2))/(3+(-3))

I'm not sure exactly how this relates to the original question, but it's true that there are conditions under which the step you asked about is not valid; they don't seem to have stated those conditions, so what they say is not universally valid. (Cubist has mentioned another condition, involving signs.)

In your example here, a/b = c/d is equal to (a+b)/(c+d) when all three ratios are defined; using my approach to prove this, letting a = kb and c = kd we find that (a+b)/(c+d) = ([b(k+1)]/[d(k+1)], which equals b/d as long as k is not -1 -- that is, only when the two given ratios are not -1 as they are in your counterexample. So that condition should be stated.

 

[inanisumeet]

I'm not sure exactly how this relates to the original question, but it's true that there are conditions under which the step you asked about is not valid; they don't seem to have stated those conditions, so what they say is not universally valid. (Cubist has mentioned another condition, involving signs.)

In your example here, a/b = c/d is equal to (a+b)/(c+d) when all three ratios are defined; using my approach to prove this, letting a = kb and c = kd we find that (a+b)/(c+d) = ([b(k+1)]/[d(k+1)], which equals b/d as long as k is not -1 -- that is, only when the two given ratios are not -1 as they are in your counterexample. So that condition should be stated.

yes , i agree.
thanks .