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Chapter 10: Problem 213

Solve the equation: \(\sqrt{\mathrm{x}}=7+\sqrt{\mathrm{x}}-7\)

Short Answer

Expert verified
The given equation is \( \sqrt{x} = 7 + \sqrt{x} - 7 \). Since both sides of the equation are identical, the equation is true for all non-negative values of x. Therefore, the solution is x 鈮� 0, which means x can be any real number greater than or equal to 0.

Step by step solution

01

Simplify the equation

Rewrite the equation while simplifying both sides to isolate the square root expressions. In this case, we simply combine the terms on the right-hand side: \( \sqrt{x} = 7 + \sqrt{x} - 7 \)
02

Cancel constant terms

Notice that the constant terms +7 and -7 on the right-hand side can cancel each other: \( \sqrt{x} = \sqrt{x} \)
03

Compare both sides of the equation

At this point, we recognize that both sides of the equation are already identical. This means that the equation is true for any non-negative value of x (since x must be non-negative for the square root to be defined). So, all non-negative values of x are solutions to the equation. In other words, x can be any real number greater than or equal to 0.

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Key Concepts

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

Isolating Square Roots
Understanding how to isolate square roots in an equation is a fundamental skill in algebra. It involves manipulating the equation so that the square root expression stands alone on one side. This is typically the first step when solving square root equations because it simplifies the equation, making it easier to work with. Here's the process:

  • Rewrite the equation in a way that identifies and separates the square root term.
  • Perform any simplifications necessary such as combining like terms or distributing.
  • Move all non-square root terms to the opposite side of the equation by adding or subtracting them.

Once the square root is isolated, you're in a good position to further solve the equation, which often involves squaring both sides to eliminate the square root.
Cancelling Constant Terms
Cancelling constant terms is another pivotal technique in algebra that helps simplify equations. Constant terms are numbers without variables. When these constants appear on both sides of the equation, they may be eliminated, which simplifies the equation. Here's how you can do it:

  • Identify constant terms on both sides of the equation.
  • If they are additive inverses of each other (one is positive, the other is negative), they can be cancelled out by adding or subtracting them from both sides.
  • This process effectively removes clutter from the equation, allowing you to focus on the variable terms that are crucial for finding the solution.

After cancelling out constants, the equation often becomes much easier to solve or may directly reveal the solution.
Comparing Sides of an Equation
Comparing sides of an equation is a vital step in verifying the solution to an equation, particularly after simplifying terms and isolating variables. It involves the following:

  • Checking if both sides of the equation are identical in terms of variable expressions.
  • If they are, the equation is true for all values that satisfy the expressions on both sides.
  • In such cases, either there are infinitely many solutions or the equation is an identity, meaning it's true no matter what the value of the variable is, provided it's within the domain of the expression.

Comparing both sides also helps in identifying when there are no solutions, typically when you end up with a statement that is always false, like a non-zero constant equalling zero.

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