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Chapter 1: Problem 24

Solve each radical equation. Check all proposed solutions. $$\sqrt{x+5}-\sqrt{x-3}-2$$

Short Answer

Expert verified
After conducting these steps, solve the quadratic equation and verify each solution by substituting it back into the original equation

Step by step solution

01

Isolate one radical

We start by isolating \(\sqrt{x+5}\) on one side of the equation: \(\sqrt{x+5} = \sqrt{x-3} +2\)
02

Square both sides

Squaring both sides will get rid of the square roots: \((\sqrt{x+5})^2 = (\sqrt{x-3} +2)^2\), Simplify to get \(x+5 = x-3+4\sqrt{x-3}+4\)
03

Solve for \(x\)

Rearranging the equation results in \(4\sqrt{x-3} = -x+6\). Squaring both sides again, we get \(16(x -3) = x^2 -12x +36\)
04

Simplify and solve the quadratic equation

Rearrange to form a quadratic equation in standard form \(x^2 -12x -16x +36 = 0\). Solve this quadratic equation using zero product property or Quadratic formula
05

Check each solution

Substitute each solution into the original equation to see if it satisfies the equation. If a solution does not satisfy the equation, then it is an extraneous solution and should be discarded

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate Radical
When solving radical equations, your first goal is to cleanly separate one of the radical terms on one side of the equation. This process is known as isolating the radical. By performing this step, you can simplify the equation and make it easier to eliminate the radical term in future steps.
  • In the given problem, start by moving all other terms to one side, leaving \( \sqrt{x+5} \) by itself.
  • This results in the equation \( \sqrt{x+5} = \sqrt{x-3} +2 \).
This allows us to address the isolated radical effectively because there are fewer distractions from other terms.
Square Both Sides
Once the radical is isolated, removing it requires that you square both sides of the equation. This step involves raising each side of the equation to the power of 2, effectively cancelling out the square root.
  • In our example, you square \( (\sqrt{x+5})^2 = (\sqrt{x-3} +2)^2 \).
  • Squaring the left side yields \( x+5 \), while the right side uses the expansion of \( (a+b)^2 = a^2 + 2ab + b^2 \).
After expanding and simplifying, you reveal a new equation without square roots, effectively allowing further reduction and simplification.
Quadratic Equation
With the square roots eliminated, the next step typically involves dealing with a quadratic equation. Quadratic equations take the form \( ax^2 + bx + c = 0 \) and can be solved using various methods.
  • After rearranging and simplifying the equation, you end up with \( x^2 -28x + 84 = 0 \).
  • Use techniques like the quadratic formula, factoring, or completing the square to find potential solutions.
These solutions are crucial but remember that the process doesn't end here because the solutions need to be verified in the context of the initial radical equation.
Extraneous Solution
An important final step in solving radical equations is checking each solution to see if it satisfies the original equation. Sometimes, squaring both sides of an equation introduces a wider set of solutions, known as extraneous solutions.
  • A potential solution might be a valid mathematical solution to the quadratic form but invalid in the original equation.
  • Substituting back into the original equation determines if a solution is genuine or extraneous.
If a solution doesn't hold true in the initial setup, disregard it. By doing so, you ensure that your final answer accurately reflects valid solutions to the original radical equation.

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Most popular questions from this chapter

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