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Chapter 8: Problem 22

Solve each equation. $$ x=\sqrt{x^{2}-2 x-6} $$

Short Answer

Expert verified
There are no solutions for this equation because \(x = -3\) does not satisfy the original equation.

Step by step solution

01

- Square both sides

To eliminate the square root, square both sides of the equation: \(x = \sqrt{x^2 - 2x - 6}\) becomes \(x^2 = x^2 - 2x - 6\)
02

- Simplify the equation

After squaring both sides, simplify the equation by moving all terms to one side: \(x^2 = x^2 - 2x - 6\) Subtract \(x^2\) from both sides: \(0 = -2x - 6\)
03

- Solve for x

Solve the linear equation for \(x\): \(0 = -2x - 6\) Add 6 to both sides: \(6 = -2x\) Divide both sides by -2: \(x = -3\)
04

- Verify the solution

Check the solution by substituting \(x = -3\) back into the original equation: \(-3 = \sqrt{(-3)^2 - 2(-3) - 6}\) Simplify inside the square root: \(-3 = \sqrt{9 + 6 - 6}\) \(-3 = \sqrt{9}\) Since \(\sqrt{9}\) is 3, not -3, the solution \(x = -3\) does not satisfy the original equation.

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Key Concepts

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

square both sides
Squaring both sides of an equation can help to eliminate a square root. This technique is often used in algebra to simplify and solve equations that involve radicals. For the equation given, we start by squaring both sides:
$$ x = \sqrt{x^{2}-2x-6} $$
Squaring both sides, we get:
$$ x^{2} = (\sqrt{x^{2}-2x-6})^{2} $$
This removes the square root, simplifying the equation to:
$$ x^{2} = x^{2}-2x-6 $$
Be careful when squaring both sides鈥攊t can introduce extraneous solutions. Therefore, always verify the solution afterward.
simplify equation
Once the equation is squared, the next step is to simplify it. Simplification makes it easier to solve. After squaring, we had:
$$ x^{2} = x^{2} - 2x - 6 $$
To simplify it further, move all terms to one side of the equation by subtracting \( x^{2} \) from both sides:
$$ x^{2} - x^{2} = x^{2} - 2x - 6 - x^{2} $$
Simplifying, we get:
$$ 0 = -2x - 6 $$
This is now a simpler equation that we can solve for \( x \).
verify solution
Verification is crucial to ensure the solution is correct, especially after operations like squaring both sides. The solution we found was \( x = -3 \). To verify, substitute it back into the original equation:
$$ x = \sqrt{x^{2}-2x-6} $$
Substituting \( x = -3 \):
$$ -3 = \sqrt{(-3)^{2} - 2(-3) - 6} $$
Simplifying inside the square root:
$$ -3 = \sqrt{9 + 6 - 6} $$
$$ -3 = \sqrt{9} $$
Notice \( \sqrt{9} = 3 \), not -3. Therefore, \( x = -3 \) doesn't work, and we have to consider if there might be any extraneous solutions introduced by squaring.
linear equation
Linear equations are the simplest type of algebraic equations, of the form \( ax + b = 0 \). They involve only constants and a single variable. After simplifying our equation:
$$ 0 = -2x - 6 $$
We solve it like a linear equation:
  • Add 6 to both sides: \( 6 = -2x \)
  • Divide both sides by -2: \( x = -3 \)
When solving, always perform the same operation to both sides to keep the equation balanced. Linear equations typically have one solution, provided the coefficients are non-zero.

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