Comparing rational and irrational parts For rational a,b,c,d and irrational n: a+b n=c+d n 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 3=d+4 3 with c and d rational, then d=5 and c=4. Equations with an unknown inside a root Isolate the radical, square both sides, then check every root in the original equation. **Working** | **Check in the original equation** sqrt(2x+5)=x+1 2x+5=(x+1)2 2x+5=x2+2x+1 x2-4=0 x=2 or x=-2 | x=2: sqrt(9)=3, x+1=3 x=-2: sqrt(1)=1, x+1=-1 x=-2 rejected x=2

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