What value of x satisfies the surd equation \sqrt{x + 7} - \sqrt{x} = 1? — JAMB UTME

What value of $x$ satisfies the surd equation $\sqrt{x + 7} - \sqrt{x} = 1$ ?

The value of

x

that satisfies the equation is

9

By rearranging the equation to

\sqrt{x+7} = \sqrt{x} + 1

and squaring both sides, we get

x + 7 = x + 2\sqrt{x} + 1

. Subtracting

x + 1

from both sides gives

6 = 2\sqrt{x}

, which yields

\sqrt{x} = 3

. Squaring both sides produces

x = 9

, which correctly satisfies the original equation.

Isolate one of the radical terms on one side of the equation.

\sqrt{x + 7} = \sqrt{x} + 1

Isolating a square root allows squaring both sides to eliminate the outer radical.

Square both sides of the equation.

x + 7 = (\sqrt{x} + 1)^2 = x + 2\sqrt{x} + 1

Expanding the right-hand side using

(a+b)^2 = a^2 + 2ab + b^2

removes the radical from the left side.

Subtract

x

and

1

from both sides to isolate the remaining radical term.

6 = 2\sqrt{x} \implies \sqrt{x} = 3

Simplifying the linear terms leaves a simple square root equation.

Square both sides to find

x

x = 3^2 = 9

Squaring

\sqrt{x}

isolates

x

completely.

Solving Surd Equations by Isolating Radicals and Squaring

Question