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Re: If a - b = 16 [#permalink]
avdesh0490 wrote:
but for substitution method it should be mentioned that a and b are integers:

Keeping this in mind write all the numbers which sums equal to 8 i.e 1+7,2+6,3+5,4+4
Now look for the difference of squares which is equal to 16 , 25-9 = 16.
Numbers are 25,9 so the ans is 15



I didn't notice this. Can you write down your explanation thoroughly?
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Re: If a - b = 16 [#permalink]
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a and b are perfect squares as the addition of their square roots equal to an integer (8). In this case, only a=25 and b=9 match the conditions. So, answer is 15 ( after some calculation, right ?
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Re: If a - b = 16 [#permalink]
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\(\sqrt{a} + \sqrt{b} =8\)

For a under square root and b as well to give 8 a and b must be perfect squares such as \(\sqrt{16} + \sqrt{16} = 4+4=8\)

However, a-b=16 which seems to contradict the statement of before.

BUT this gives us a clue: a and b must be a perfect square. As such, for a-b =16 follow that a must be 25 and b must be 9, which are perfect squares.

\(25 \times 9= 225\) which is a perfect square and is equal to 15

E is the answer
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Re: If a - b = 16 [#permalink]
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msdjaw wrote:
If \(a - b = 16\) and \(\sqrt{a} + \sqrt{b} = 8\), what is the value of \(\sqrt{ab}\) ?

A. 2

B. 4

C. 8

D. 12

E. 15



GIVEN: √a + √b = 8


GIVEN: a - b = 16
We can think of a - b as a difference of squares, since (√a)² = a, and (√b)² = b

So, take a - b = 16
Rewrite as: (√a + √b)(√a - √b) = 16
Substitute to get: (8)(√a - √b) = 16
Divide both sides by 8 to get: √a - √b = 2

We now have:
√a + √b = 8
√a - √b = 2
ADD the equations to get: 2√a = 10, which means √a = 5

Take:
√a + √b = 8
√a - √b = 2
SUBTRACT the bottom equation from the top equation to get: 2√b = 6, which means √b = 3

So, √(ab) = (√a)(√b) = (5)(3) = 15

Answer: E

Cheers,
Brent
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Re: If a - b = 16 [#permalink]
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msdjaw wrote:
If \(a - b = 16\) and \(\sqrt{a} + \sqrt{b} = 8\), what is the value of \(\sqrt{ab}\) ?

A. 2

B. 4

C. 8

D. 12

E. 15

Hello!

I have the answer to this question in my book using other methods. However, whenever I try to solve this question using the substitution method, my solution is not the same as what my book says. Would you please be kind enough to solve this question for me using the SUBSTITUTION method?

I know there are easier methods as well but I need to know what mistake I am making by using the substitution method.





Here is an algebraic solution to the problem.
First of all, we need to simplify the question.
Let's say: The sum of two numbers/integers is equal to 8 and the difference between the square of the two numbers/integers is equal to 16. Find the two numbers/integers and calculate the value of the square root of the product of the two numbers/integers.

Let's \(\sqrt{a}\) be the first number/integer and \(\sqrt{b}\) the second number/integer

\(\sqrt{a}+\sqrt{b}=8\)
\(\sqrt{a}^2-\sqrt{b}^2=16\)
\((\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) =16\)
\(8(\sqrt{a}-\sqrt{b})=16\)
\(\sqrt{a}-\sqrt{b}=2\)

Then solve the two equations
\(\sqrt{a}+\sqrt{b}=8\)
\(\sqrt{a}-\sqrt{b}=2\)
\(2\sqrt{a}=10\)
\(\sqrt{a}=5\)
\(\sqrt{b}=3\)

Thus, \(a=25\)
and \(b=9\)
\(\sqrt{ab}=\sqrt{25*9}\)

The answer is 15.
I hope that the formula will work
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Re: If a - b = 16 [#permalink]
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Re: If a - b = 16 [#permalink]
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