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Chapter 10: Problem 11

Solve the given equation. $$\sqrt{x}+3=10$$

Short Answer

Expert verified
The solution is x = 49.

Step by step solution

01

Isolate the square root term

To solve the equation \(\root{x} + 3 = 10\), the first step is to isolate the term containing the square root. You can do this by subtracting 3 from both sides of the equation.\[\sqrt{x} + 3 - 3 = 10 - 3\] Simplifies to: \[\root{x} = 7\]
02

Square both sides

Now that the square root term is isolated, square both sides of the equation to eliminate the square root. \(\sqrt{x} = 7\) becomes: \[ (\root{x})^2 = 7^2 \] Which simplifies to: \[ x = 49 \]
03

Verify the solution

Always verify your solution by substituting it back into the original equation. Substitute \x = 49\ into \(\root{x} + 3 = 10\):\ \(\root{49} + 3 = 10\) which simplifies to\ \(7 + 3 = 10\) which is a true statement, confirming that the solution is correct.

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Key Concepts

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

Square Root
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 49 is 7, because 7 * 7 = 49. In mathematical notation, we use the radical symbol \( \sqrt{} \) to denote the square root. Solving equations involving square roots often involves isolating the square root term and then squaring both sides to remove the square root. Key concepts to remember about square roots:
  • The square root of a number is always non-negative.
  • Each positive number has two square roots: one positive (principal square root) and one negative.
  • The square root function is the inverse of the squaring function.

Isolate the Variable
Before you can solve an equation involving a square root, you need to isolate the square root term. Isolating the variable means getting it by itself on one side of the equation. In the given problem, the equation is \( \sqrt{x} + 3 = 10 \). Start by isolating \( \sqrt{x} \) by subtracting 3 from both sides:

\[ \sqrt{x} + 3 - 3 = 10 - 3 \] Simplifying this gives us
\[ \sqrt{x} = 7 \]Here are some general tips to effectively isolate the variable:
  • Always perform the same operation on both sides of the equation.
  • Simplify step by step to keep track of each change.
  • Double-check each step to ensure accuracy.
By isolating the variable, you make it easier to solve the remaining equation.
Verify Solutions
It's crucial to verify your solution after solving an equation. This ensures that the solution you found is correct. To verify a solution, substitute it back into the original equation and check if both sides of the equation are equal. For the problem \( \sqrt{x} + 3 = 10 \), we found \( x = 49 \). We then substitute x into the original equation:

\[ \sqrt{49} + 3 = 10 \] Simplifying this, we get:
\[ 7 + 3 = 10 \] which is a true statement. This confirms that our solution, \( x = 49 \), is correct.Remember these points for verification:
  • Always use the original equation for substitution.
  • Check each part of your equation during substitution.
  • Ensure both sides of the equation match after substitution.
Validating your answer this way ensures your solution is accurate and error-free.

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Most popular questions from this chapter

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