Examples of solved irrational equations

Two worked examples that show the method in action: in the first the equation has no solutions, in the second one of the two solutions is extraneous and must be discarded.

Example 1 — No solution

$x - 2\sqrt{x+1} + 3 = 0$.

Isolate: $-2\sqrt{x+1} = -3-x \implies \sqrt{x+1} = \dfrac{x+3}{2}$.

Domain of existence: $x+1\ge 0 \implies x\ge -1$.

Sign-agreement condition: $\dfrac{x+3}{2}\ge 0 \implies x\ge -3$.   Total domain of existence: $x\ge -1$.

Square: $x+1 = \frac{(x+3)^2}{4} \implies 4x+4 = x^2+6x+9 \implies x^2+2x+5=0$ $\Delta = 4-20 = -16 < 0$: no solution.

Example 2 — An extraneous solution to discard

$\sqrt{8-x}-2x+6=0$.

Isolate: $\sqrt{8-x}=2x-6$.

Domain of existence: $8-x\ge 0 \implies x\le 8$.

Sign-agreement condition: $2x-6\ge 0 \implies x\ge 3$.   Total domain of existence: $3\le x\le 8$.

Square: $8-x = (2x-6)^2 = 4x^2-24x+36 \implies 4x^2-23x+28=0$ $\Delta = 529-448 = 81 = 9^2$ $x_A = \frac{23+9}{8} = 4 \quad(\checkmark\ \text{in } [3,8]), \qquad x_B = \frac{23-9}{8} = \frac{7}{4} \quad(\tfrac{7}{4}<3,\ \text{scartata})$

Check: $\sqrt{8-4}-2(4)+6 = 2-8+6 = 0$ ✓.

Solution: $\boxed{x=4}$.

Links

Topics: Radicals
Concepts: Irrational equation · Extraneous solution
Methods: Irrational equation squaring
Skills: Solving equations