Question

if √2+√7 /√2-√7 =a+b √14,find a, b​

Answer 1

Step-by-step explanation:

Oh boy, this was a fun one.

(a−b)2 expands to a2–2ab+b2, which equals 7.

We know that ab equals 14, so substitute it into the above equation.

a2–28+b2=7

a2+b2=35

Now for some creativity. a2+b2 is the “newbie” answer to the expansion of (a+b)2: i.e. with the midsection missed out:

(a+b)2=a2+2ab+b2

Now we know that a2+b2=35 and 2ab=28, so add the two together to get (a+b)2=a2+2ab+b2=35+28=63

Therefore (a+b)2=63. Now to find out a and b’s vaaaalyeeeews. By rooting both sides we can see that a+b=±37–√.

Notice that having both a+b and ab is the “product/sum” stage of a basic quadratic factorisation: (x+a)(x+b)=x2+(a+b)x+ab

So now we can use the quadratic formula to filter out our solutions.

x=∓37–√±(37–√)2−(4×1×14)−−−−−−−−−−−−−−−−−√2

x=∓37–√±63–56−−−−−√2

x=∓37–√±7–√2

You’ll end up with the following solutions:

x=7–√,27–√

and

x=−7–√,−27–√

But b>0, so...

Our final answer: a and b are 7–√ and 27–√. In either order.

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