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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