How do I get the 9g^2+49h^2 to work with g^2+h^2?

senseimichael

senseimichael

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If g^2+h^2=172 and 4hg=408, find the value of (3g-7h)^2.

Expanding the expression gives me 9g^2- 42hg + 49h^2 [edited]. Getting the 42hg solved is easy enough, but how on earth do I solve the 9g^2+49h^2?
 
Last edited by a moderator:
This is a system of simultaneous equations. You can solve it for numeric values of g and h using the most general way to solve such systems: substitution. Once you you know g and h, finding (3g - 7h)^2 is a cinch.

[MATH]4hg = 408 \implies h = \dfrac{104}{g} \implies h^2 = \dfrac{104^2}{g^2}.[/MATH]
[MATH]\therefore 9g^2 + h^2 = 172 \implies 9g^2 + \dfrac{104^2}{g^2} = 172 \implies 9g^4 + 104^2 = 172g^2 \implies[/MATH]
[MATH]9g^4 - 172g^2 + 104^2 = 0.[/MATH]
Now what?
 
senseimichael said:
If g^2+h^2=172 and 4hg=408, find the value of (3g-7h)^2.

Expanding the expression gives me 9g^2+42hg+49h^2. Getting the 42hg solved is easy enough, but how on earth do I solve the 9g^2+49h^2?
Click to expand...
\(\displaystyle (3g-7h)^2 \neq 9g^2+42hg+49h^2\)
 
JeffM said:
This is a system of simultaneous equations. You can solve it for numeric values of g and h using the most general way to solve such systems: substitution. Once you you know g and h, finding (3g - 7h)^2 is a cinch.

[MATH]4hg = 408 \implies h = \dfrac{104}{g} \implies h^2 = \dfrac{104^2}{g^2}.[/MATH]
[MATH]\therefore 9g^2 + h^2 = 172 \implies 9g^2 + \dfrac{104^2}{g^2} = 172 \implies 9g^4 + 104^2 = 172g^2 \implies[/MATH]
[MATH]9g^4 - 172g^2 + 104^2 = 0.[/MATH]
Now what?
Click to expand...
To OP: JeffM meant to say \(\displaystyle h = \dfrac{102}{g}\)
 
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I think there's a bit of an easier way to go about this.

[MATH] (g+h)^2 = g^2 + 2gh + h^2 = 172 + 204 = 376\\ g+h = \pm \sqrt{376} = 2\sqrt{94}\\ gh = 102\\ g + \dfrac{102}{g} = 2\sqrt{94}\\ g^2 - 2\sqrt{94}g + 102 = 0\\ \text{Now we can apply the quadratic formula}\\ g = \dfrac{2\sqrt{94} \pm \sqrt{376-408}}{2} = \sqrt{94} \pm i2\sqrt{2}\\ h = \sqrt{94} \mp i 2\sqrt{2} = g^*\\ (3g-7h)^2 = 64(11 \pm i5\sqrt{47})\\ \text{which is why I suspect a typo somewhere} [/MATH]
 
I am supposed to use the fact that (a-b)2=a2-2ab+b2 to work out the answer. I am already given a multiple of ab, and also a2+b2. I just cannot find a way to manipulate that 3g-7h into multiples of a2+b2. Knowing the paper (and the setter of the paper) I am working with, I really *do* suspect there is a typo in this question.
 
senseimichael said:
Oops. It should be 9g2-42gh+49h2, my bad.

And I don't suppose I can edit my original post... :(
Click to expand...
Now you have been given several ways to solve the problem with typo. Please show us your work with "corrected" problem.
 
senseimichael said:
I couldn't. The solutions given could not bring me to the multiples of g2+h2, which is needed to solve the problem.
Click to expand...
Please show us what you have tried - even if you know that you are "incorrect".

That will give us an idea about where you are going wrong and how we could guide you to the answer.
 
Subhotosh Khan said:
Please show us what you have tried - even if you know that you are "incorrect".

That will give us an idea about where you are going wrong and how we could guide you to the answer.
Click to expand...

I have left with me 9g2+49h2. I tried taking out the 9, giving me 9(g2+(49/9)h2). That is most definitely NOT a multiple of g2+h2 (which itself is 172, a nice value), and so I am stuck. I am indeed inclined to believe that the question as presented cannot be solved and so there could have been an error in the question originally.
 
senseimichael said:
If g^2+h^2=172 and 4hg=408, find the value of (3g-7h)^2.

Expanding the expression gives me 9g^2+42hg+49h^2. Getting the 42hg solved is easy enough, but how on earth do I solve the 9g^2+49h^2?
Click to expand...
If I were to do this problem, I would first solve for 'g' and 'h' in the following way:

g^2 + h^2 + 2hg ...\(\displaystyle \to \) ... (g + h)^2 = ??? ...\(\displaystyle \to \) ... g + h = ???

g^2 + h^2 - 2hg ...\(\displaystyle \to \) ... (g - h)^2 = ??? ...\(\displaystyle \to \) ... g - h = ???

From above g = ? and h = ? and eventually (3g - 7h)^2 = ?

This is same as response #6.

There might be "multiple" answers.
 
Thank you so much, guys, for all the help. I am going to assume a typo in the question and shall raise this up to the paper setter.
 
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