If \sqrt{x + 2} = x, what is the value of x - 1? — SAT

If $\sqrt{x + 2} = x$ , what is the value of $x - 1$ ?

The value of

x - 1

1

The valid solution to the radical equation is

x = 2

. Substituting this into the expression

x - 1

yields

2 - 1 = 1

Square both sides of the equation to eliminate the radical.

x + 2 = x^2

Squaring both sides removes the square root, allowing us to solve the resulting equation.

Rewrite the equation in standard quadratic form.

x^2 - x - 2 = 0

Moving all terms to one side sets the quadratic equation to zero, which is necessary for factoring.

Factor the quadratic equation.

(x - 2)(x + 1) = 0

, which gives the potential solutions

x = 2

and

x = -1

Factoring allows us to find the roots of the quadratic equation.

Check both potential solutions in the original equation to identify any extraneous solutions.

For

x = 2

\sqrt{2 + 2} = 2

is true. For

x = -1

\sqrt{-1 + 2} = -1 \implies 1 = -1

is false.

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical equation.

Calculate the value of

x - 1

using the valid solution.

2 - 1 = 1

The question asks for the value of the expression

x - 1

rather than just the variable

x

Radical Equations and Extraneous Solutions

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