8.4 Practice Problems
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Solution. Note first that both roots require \(x\geq 1\). Squaring both sides, \[x+4=\left (\sqrt {x-1}+1\right )^{2}=(x-1)+2\sqrt {x-1}+1=x+2\sqrt {x-1},\] so \(4=2\sqrt {x-1}\) and \(\sqrt {x-1}=2\). Squaring again, \(x-1=4\) and \(x=5\).
Check in the original: \(\sqrt {9}=3\) and \(\sqrt {4}+1=3\). The two agree, so \(x=5\) is genuine.
Note 8.13. Squaring an equation with two separate roots usually leaves one of them behind, as here — the first squaring cleared \(\sqrt {x+4}\) and left \(\sqrt {x-1}\), which needed a second. Isolate one root on its own side before squaring, or the cross term will be worse than what you started with.
Only one candidate appeared this time, so nothing had to be discarded, but the check is still required: squaring can only ever add solutions, never lose them, so every candidate must be tested against the original.
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