Solving the Equation: The Square Root of the Sum of Twice a Number and 3 is 6
In this article, we will walk through a detailed step-by-step process to solve a mathematical problem wherein the square root of the sum of twice a number and 3 is equal to 6. We will break down the equation, solve it, and discuss the importance of each step in the problem-solving process. This guide is designed to be accessible to beginners while offering valuable insights for more experienced learners.
Understanding the Problem
The problem at hand is: The square root of the sum of twice a number and 3 is equal to 6. Let's denote the unknown number as ( x ).
The Equation
According to the problem, we can write the equation as:
sqrt{2x 3} 6
Solving the Equation Step-by-Step
To solve for ( x ), we need to eliminate the square root first. We can achieve this by squaring both sides of the equation.
Squaring Both Sides
Squaring both sides, we get:
sqrt{2x 3}^2 6^2
This simplifies to:
2x 3 36
Isolating the Variable
To isolate the variable ( x ), we need to follow a series of algebraic manipulations. First, we will subtract 3 from both sides of the equation.
2x 3 - 3 36 - 3
This simplifies to:
2x 33
Next, we divide both sides of the equation by 2 to solve for ( x ).
2x / 2 33 / 2
This simplifies to:
x 33 / 2 16.5
Therefore, the number is ( boxed{16.5} ).
Conclusion and Practice
This problem demonstrates the importance of understanding and applying algebraic principles such as isolating variables and manipulating equations. Solving equations through such steps is crucial in various fields, including physics, engineering, and data science.
To further reinforce the understanding of this concept, consider the following practice questions:
Practice Questions
What is the number that satisfies the equation ( sqrt{2n - 3} 6 )? Find the number ( x ) if the square root of the sum of twice the number and 5 is 8. Solve for the number ( y ) if ( sqrt{2y 4} 7 ).By practicing these types of problems, you can improve your algebraic problem-solving skills and deepen your understanding of mathematical concepts.