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# A funny algebra puzzle!

a and b are real numbers

We have the following expressions:

a² - b² = 9

a.b = 3

Question: what are the value(s) for a and b ?

Good luck! And try to be among the first ones!

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[edited by: emassa at 2:37 PM (GMT -5) on 20 Feb 2023]
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• OK  it's time to publish the official (and right) solution! There are well real solutions for these equations!

From the second equation we can write a = 3/b

Injecting it in the first equation: 9/b² ‐ b² = 9

Or 9 ‐b4 = 9b² or b4 + 9b² ‐ 9 = 0 Let’s pose b²=y

We have then y² + 9y – 9 = 0

And hence y = [‐9 +/‐ (81 +36)1/2]/2 = [‐9+/‐1171/2]/2

Or y = 0.908 and ‐9.908

Since y = b² must be positive, the only value for y is 0.908

Therefore b = +/‐ 0.9531 and a = 3/b = +/‐ 3.1476

There are 2 sets of possible solutions:

Values for a, b are thus :

‐3.1476 and ‐0.9531

And +3.1476 and +0.9531

Big applause to our 4 first winners:

1. Gaetano PIARINO, Key Account Manager - ADI - Milan, Italy

2. Ricardo HABRE, Field Application Engineer - ADI - Farmington (MI), USA

3. Mark CEE, Senior Engineer, Systems Integration Engineering – ADI – Cavite, Philippines

4. Martin WALKER, Product Marketing Lead Engineer – ADI – Newbury, UK

Be ready for the next coming quiz!

• OK  it's time to publish the official (and right) solution! There are well real solutions for these equations!

From the second equation we can write a = 3/b

Injecting it in the first equation: 9/b² ‐ b² = 9

Or 9 ‐b4 = 9b² or b4 + 9b² ‐ 9 = 0 Let’s pose b²=y

We have then y² + 9y – 9 = 0

And hence y = [‐9 +/‐ (81 +36)1/2]/2 = [‐9+/‐1171/2]/2

Or y = 0.908 and ‐9.908

Since y = b² must be positive, the only value for y is 0.908

Therefore b = +/‐ 0.9531 and a = 3/b = +/‐ 3.1476

There are 2 sets of possible solutions:

Values for a, b are thus :

‐3.1476 and ‐0.9531

And +3.1476 and +0.9531

Big applause to our 4 first winners:

1. Gaetano PIARINO, Key Account Manager - ADI - Milan, Italy

2. Ricardo HABRE, Field Application Engineer - ADI - Farmington (MI), USA

3. Mark CEE, Senior Engineer, Systems Integration Engineering – ADI – Cavite, Philippines

4. Martin WALKER, Product Marketing Lead Engineer – ADI – Newbury, UK

Be ready for the next coming quiz!

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