Problem 41 - Precalculus : Building Concepts and Connections

Question

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt{x+1}+2=x$$

Step-by-Step Solution

Answer

The real solution to the radical equation is \(x = 3\).

Step 1: Isolate the radical expression

Subtract 2 from both sides of the equation to make the square root term by itself on one side: \(\sqrt{x+1} = x-2\)

Step 2: Square both sides

Square both sides of the equation to eliminate the square root. Remember the formula \((a-b)^2 = a^2 - 2ab + b^2\). This yields \(x+1 = (x-2)^2 = x^2 - 4x + 4\)

Step 3: Simplify to quadratic equation

Transform the equation into a quadratic equation. To do it, bring all terms to one side: \(0 = x^2 - 4x - x + 4 - 1= x^2 - 5x + 3\)

Step 4: Solve the quadratic equation

Solve this quadratic equation, by looking for factors of 3 that sum to 5. The factors are 1 and 3, hence the roots are 1 and 3, or \(x = 1, x = 3\).

Step 5: Check the solutions

Check both solutions in the original equation. For \(x = 1\): \(\sqrt{1+1}+2 = 1+2 =3\) which is not equal to 1, so this is not a solution. For \(x = 3\): \(\sqrt{3+1}+2= 3\) which holds true, so 3 is a valid solution for the equation.

Key Concepts

Square Root Equations

Solving square root equations, such as ewline

Quadratic Equations

Quadratic equations are in the form ewline

Isolation of Radical Expression

The isolation of a radical expression is a key strategy in solving equations that involve roots. Essentially, this means you want to get the term with the root by itself on one side of the equation to facilitate the removal of the root. This isolation step is crucial for the subsequent steps in solving radical equations because it prepares the equation for squaring, which eliminates the radical. For example, in the step by step solution provided, the first thing we did was move the 2 from the left-hand side to the right-hand side to isolate the ewline

Checking Solutions in Algebra

When solving algebraic equations, checking your solutions in the context of the original equation is crucial to verify their correctness. This is especially important when dealing with radical equations because the act of squaring both sides can introduce extraneous solutions. These are solutions that appear valid when plugged into the modified equation but do not satisfy the original equation. After identifying potential solutions, you should substitute them back into the original radical equation to confirm that they produce a true statement.

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Problem 41

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