If √ (2x + 2) − √x =1 ,find x.

Analysis

To solve the equation $\sqrt{2x + 2} - \sqrt{x} = 1$, we need to isolate the radical terms and square both sides to eliminate the square roots.

Concept: Solving Radical Equations

When dealing with equations involving square roots, the standard procedure is:

1. Isolate one radical on one side of the equation.
2. Square both sides.
3. If a radical remains, repeat the process.
4. Always check your solutions in the original equation, as squaring can sometimes introduce "extraneous" (false) roots.

Step-by-Step Solution

Step 1: Isolate one radical. $\sqrt{2x + 2} = 1 + \sqrt{x}$

Step 2: Square both sides. $(\sqrt{2x + 2})^2 = (1 + \sqrt{x})^2$ $2x + 2 = 1 + 2\sqrt{x} + x$

Step 3: Simplify and isolate the remaining radical. Subtract $x$ and $1$ from both sides: $(2x - x) + (2 - 1) = 2\sqrt{x}$ $x + 1 = 2\sqrt{x}$

Step 4: Square both sides again to remove the final radical. $(x + 1)^2 = (2\sqrt{x})^2$ $x^2 + 2x + 1 = 4x$

Step 5: Solve the resulting quadratic equation. $x^2 + 2x - 4x + 1 = 0$ $x^2 - 2x + 1 = 0$

Factor the quadratic: $(x - 1)(x - 1) = 0$ $(x - 1)^2 = 0$ $x = 1$

Step 6: Verify the solution. Substitute $x = 1$ into the original equation: $\sqrt{2(1) + 2} - \sqrt{1} = 1$ $\sqrt{4} - 1 = 1$ $2 - 1 = 1$ $1 = 1$ (The solution is valid).

Since the quadratic equation $(x-1)^2 = 0$ yields $x=1$ as a repeated root, we say $x=1$ twice.

Conclusion

This matches our algebraic result.

Exam Tip

Whenever you see a radical equation in JAMB, always remember to check your answers. Sometimes, squaring both sides creates a value that doesn't actually work in the original equation. In this case, $x=1$ works perfectly!