Radical equations and functions Calculator

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Example

Created on 2024-06-20Asked by Avery Rivera (Solvelet student)

Solve the radical equation:

\sqrt{x + 5} = x - 1

Solution

To solve the radical equation

\sqrt{x + 5} = x - 1

: 1. Square both sides of the equation to eliminate the square root:

(\sqrt{x + 5})^2 = (x - 1)^2.

2. Simplify both sides:

x + 5 = x^2 - 2x + 1.

3. Rearrange the equation to form a quadratic equation:

x^2 - 3x - 4 = 0.

4. Factor the quadratic equation:

(x - 4)(x + 1) = 0.

5. Solve for

x

x - 4 = 0 \quad \Rightarrow \quad x = 4,

x + 1 = 0 \quad \Rightarrow \quad x = -1.

6. Check the solutions in the original equation: - For

x = 4

\sqrt{4 + 5} = 4 - 1 \quad \Rightarrow \quad 3 = 3 \quad \text{(valid)}.

- For

x = -1

\sqrt{-1 + 5} = -1 - 1 \quad \Rightarrow \quad \sqrt{4} = -2 \quad \Rightarrow \quad 2 \neq -2 \quad \text{(invalid)}.

Therefore, the solution to the equation is

x = 4

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Avery Young on Solvelet

1. Solve the equation

\sqrt{2x - 5} = 3

.2. Graph the function

f(x) = \sqrt{x^2 - 4}

and determine its domain and range.

DefinitionRadical equations contain radicals with unknowns in the radicand The solution to those kinds of equations will typically involve isolating the radical on one side, and then simply squaring (to get rid of the square root) both sides of the equation. These are some questions which can be easily solved by squaring also, e.g; x+2=3 -> x+2=9 so, x=7.

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