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The number \(a+\sqrt{b}\)  and its radical conjugate have a sum of \(-4\)  and a product of \(1\) . Find a+b.

 Aug 4, 2022
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\(a+\sqrt{b}\), its conjugate: \(a-\sqrt{b}\)

If their sum is -4:

\(a+\sqrt{b}+a-\sqrt{b}=-4 \implies 2a=-4 \implies a=-2\)

If their product is 1:

\((2+\sqrt{b})(2-\sqrt{b})=1 \iff 4-2\sqrt{b}+2\sqrt{b}-b=1\)

So:

\(4-b=1 \implies b=3\)

Therefore, \(a+b=-2+3=1\)

 Aug 4, 2022

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