Problem 38 - Algebra 2

Question

Solve. Check for extraneous solutions. \(\sqrt{x^{2}+3}=x+1\)

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{x^{2}+3}=x+1\) is \(x = 0.5\).

Step 1: Square Both Sides

To start solving this equation, the first step is to remove the square root. To do that, square both sides of the equation. This will result in \(x^{2} + 3 = (x + 1)^{2}\). Expand the right side of the equation, which gives \(x^{2} + 3 = x^{2} + 2x + 1\).

Step 2: Simplify the Equation

Subtract \(x^{2}\) from both sides to remove the quadratic term and simplify further, resulting in \(2 = 2x + 1\). Now, isolate \(x\) by subtracting 1 from both sides, yielding \(1 = 2x\), and finally divide both sides by 2, which gives \(x = 0.5\).

Step 3: Check for Extraneous Solutions

Substitute \(x = 0.5\) back into the original equation \(\sqrt{x^{2} + 3} = x + 1\) to check if it's a valid solution or if it's extraneous. Upon substitution, we find \(\sqrt{(0.5)^{2} + 3} = 0.5 + 1\). Simplifying both sides, we get \(1.5 = 1.5\), which is true, so \(x = 0.5\) is a valid solution.

Key Concepts

Solving Equations

Solving equations often involves finding the unknown value that makes the equation true. To solve an equation, you typically use operations like addition, subtraction, multiplication, or division until you isolate the variable on one side. Finding a solution means you have determined the specific value of the variable that satisfies the equation. The goal is to simplify the equation as much as possible until the variable is isolated or obvious, allowing for easy identification of its value.

• Always perform the same operation on both sides of the equation to maintain balance.
• If the equation involves variables on both sides, bring them together before isolating them.
• Simplifying complex equations step-by-step often aids in accurate and confident solving.

This procedure applies to various types of equations, each with its unique steps and methods. With this in mind, understanding the fundamental principles of equation solving serves as the bedrock for tackling any equation.

Square Root Equations

Square root equations include terms that involve a square root. These types of equations can be trickier. To solve a square root equation, like \(\sqrt{x^2 + 3} = x + 1\), the first step is to eliminate the square root by squaring both sides of the equation.
Here are a few steps that are helpful:

• Square the equation: This removes the square root and simplifies the equation. In this case, squaring both sides resulted in \(x^2 + 3 = (x + 1)^2\).
• Expand and simplify: After removing the square root, expand any squared terms and then simplify the equation. For example, \((x + 1)^2\) becomes \(x^2 + 2x + 1\).
• Proceed to solve the resulting simplified equation: Apply basic algebraic operations to isolate and find the variable.

By eliminating the square root correctly, you convert the problem into a standard algebraic equation, making it easier to find the solution.

Checking Solutions

Once you find an answer to an equation, it's important to verify it. This is especially crucial for equations with operations like square roots, where extraneous solutions can appear due to squaring both sides of an equation.
An extraneous solution is a number that appears to be a solution but doesn't satisfy the original equation.To check solutions:

• Substitute the solution back into the original equation to ensure both sides are equal.
• For the problem \(x = 0.5\), plugging back into \(\sqrt{x^2 + 3} = x + 1\) affirmed it was a valid solution after simplifying to \(1.5 = 1.5\).
• If the sides do not match, the solution is extraneous, meaning it doesn't actually solve the original problem.

This checking process helps confirm that the solution is correct and eliminates any incorrect extraneous results. It reinforces the correctness and reliability of the solution obtained.