<p class="paragraph-header">Comparing rational and irrational parts</p>
<p>For rational \(a,b,c,d\) and irrational \(\sqrt n\):</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(a+b\sqrt n=c+d\sqrt n\quad\Longrightarrow\quad a=c,\ b=d\)</p></div></div>
<p>Expand and group each side into a rational part plus a multiple of the surd, then match the two sides. For example, if&nbsp;\(5+c\sqrt3=d+4\sqrt3\) with \(c\) and \(d\) rational, then&nbsp;\(d=5\) and \(c=4\).</p>
<p class="paragraph-header">Equations with an unknown inside a root</p>
<p>Isolate the radical, square both sides, then check every root in the original equation.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 48.2143%;"><col style="width: 51.7857%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Working</strong></th><th style="text-align: center; --cell-justify: center"><strong>Check in the original equation</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\begin{aligned} \sqrt{2x+5}&=x+1 \\ 2x+5&=(x+1)^2 \\ 2x+5&=x^2+2x+1 \\ x^2-4&=0 \\ x&=2\ \text{or}\ x=-2 \end{aligned}\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>

For rational \(a,b,c,d\) and irrational \(\sqrt n\):

\(a+b\sqrt n=c+d\sqrt n\quad\Longrightarrow\quad a=c,\ b=d\)

Expand and group each side into a rational part plus a multiple of the surd, then match the two sides. For example, if \(5+c\sqrt3=d+4\sqrt3\) with \(c\) and \(d\) rational, then \(d=5\) and \(c=4\).

Isolate the radical, square both sides, then check every root in the original equation.

\(\begin{aligned} \sqrt{2x+5}&=x+1 \\ 2x+5&=(x+1)^2 \\ 2x+5&=x^2+2x+1 \\ x^2-4&=0 \\ x&=2\ \text{or}\ x=-2 \end{aligned}\)

\(\begin{aligned} x&=2:\ \sqrt{9}=3,\ x+1=3 \\ x&=-2:\ \sqrt{1}=1,\ x+1=-1 \end{aligned}\)
\(x=-2\) rejected
\(\therefore\ x=2\)

Equations Involving Surds

Equations Involving Surds