Equations with More Than One Radical

Property Sometimes we need to square both sides of an equation more than once in order to eliminate all the radicals. The general strategy is to first isolate the more complicated radical on one side of the equation. Square both sides, simplify, and then isolate the remaining radical. Finally, square both sides again to find the solution, and always check for extraneous roots.

Caution: We cannot solve a radical equation by squaring each term separately. An expression like $(\sqrt{x 7}+\sqrt{x})^2$ must be expanded as a binomial.

Examples Solve $\sqrt{x 5}+\sqrt{x}=5$. Isolate a radical: $\sqrt{x 5}=5 \sqrt{x}$. Square both sides: $x 5=(5 \sqrt{x})^2 = 25 10\sqrt{x}+x$. Isolate the remaining radical: $ 30= 10\sqrt{x}$, so $3=\sqrt{x}$. Square again: $x=9$. Solve $\sqrt{y+8} \sqrt{y}=2$. Isolate a radical: $\sqrt{y+8}=2+\sqrt{y}$. Square both sides: $y+8 = 4+4\sqrt{y}+y$. Simplify: $4=4\sqrt{y}$, so $1=\sqrt{y}$. Square again to get $y=1$. Solve $\sqrt{3x+4}=\sqrt{x 1}+3$. Square both sides: $3x+4=(\sqrt{x 1}+3)^2=x 1+6\sqrt{x 1}+9$. Simplify: $2x 4=6\sqrt{x 1}$, so $x 2=3\sqrt{x 1}$. Square again: $(x 2)^2=9(x 1)$, so $x^2 4x+4=9x 9$. This gives $x^2 13x+13=0$. This requires the quadratic formula for solutions.