We square the expression on the right-hand-side and expand
using the binomial formula
:
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(√a+√a2-b2±√a-√a2-b2)2 |
=(√a+√a2-b2)2 |
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+(√a-√a2-b2)2±2√a+√a2-b2√a-√a2-b2 |
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Since the squaring operation undoes the square roots, we
obtain the following:
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(√a+√a2-b2)2+(√a-√a2-b2)2=a+√a2-b2+a-√a2-b2=a |
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Since the product of square roots equals the square root
of the product, we have the following:
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√a+√a2-b2√a-√a2-b2 |
=√a+√a2-b2⋅a-√a2-b2 |
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=√a2-(√a2-b)24 |
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=√a2-(a2-b)4 |
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=√b4=√b2 |
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Combining what we have calculated above, we obtain
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(√a+√a2-b2±√a-√a2-b2)2=a±√b. |
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Because the square of the asserted value of the square root equals the radicand (a±√b) of the square root, and the asserted value of the square root is clearly non-negative, we have justified the validity of the formulas
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√a±√b=√a+√a2-b2±√a-√a2-b2. |
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