Solve The Following Equation: $ 10 + \sqrt{10m - 1} = 13 }$Answer { M = \text{Blank 1 $}$
Introduction
In this article, we will be solving a quadratic equation involving a square root. The equation is given as 10 + √(10m - 1) = 13, and we need to find the value of 'm'. This type of equation is commonly encountered in algebra and is a fundamental concept in mathematics.
Step 1: Isolate the Square Root
To solve the equation, we first need to isolate the square root term. We can do this by subtracting 10 from both sides of the equation.
10 + √(10m - 1) = 13
Subtracting 10 from both sides:
√(10m - 1) = 3
Step 2: Square Both Sides
Next, we need to get rid of the square root. We can do this by squaring both sides of the equation.
(√(10m - 1))^2 = 3^2
Using the property of exponents, we can simplify the left-hand side:
10m - 1 = 9
Step 3: Add 1 to Both Sides
Now, we need to isolate the term involving 'm'. We can do this by adding 1 to both sides of the equation.
10m - 1 + 1 = 9 + 1
Simplifying:
10m = 10
Step 4: Divide Both Sides by 10
Finally, we can solve for 'm' by dividing both sides of the equation by 10.
10m / 10 = 10 / 10
Simplifying:
m = 1
Conclusion
In this article, we solved the equation 10 + √(10m - 1) = 13 and found the value of 'm' to be 1. This type of equation is commonly encountered in algebra and is a fundamental concept in mathematics. By following the steps outlined above, we can solve similar equations and find the values of the variables involved.
Tips and Tricks
- When solving equations involving square roots, it's essential to isolate the square root term first.
- Squaring both sides of the equation can help eliminate the square root.
- Adding or subtracting the same value from both sides of the equation can help isolate the term involving the variable.
Real-World Applications
Solving equations involving square roots has numerous real-world applications. For example, in physics, the equation for the time it takes for an object to fall from a certain height involves a square root term. In engineering, the equation for the stress on a beam involves a square root term. By solving these types of equations, we can gain a deeper understanding of the underlying physics and engineering principles.
Common Mistakes
- Failing to isolate the square root term first.
- Not squaring both sides of the equation.
- Not adding or subtracting the same value from both sides of the equation.
Practice Problems
Try solving the following equation:
√(x + 2) = 5
What is the value of 'x'?
Answer
To solve the equation, we need to isolate the square root term first.
√(x + 2) = 5
Squaring both sides:
x + 2 = 25
Subtracting 2 from both sides:
x = 23
Therefore, the value of 'x' is 23.
Conclusion
Introduction
In our previous article, we solved the equation 10 + √(10m - 1) = 13 and found the value of 'm' to be 1. In this article, we will answer some frequently asked questions related to solving equations involving square roots.
Q: What is the first step in solving an equation involving a square root?
A: The first step in solving an equation involving a square root is to isolate the square root term. This can be done by subtracting or adding the same value from both sides of the equation.
Q: How do I eliminate the square root term from an equation?
A: To eliminate the square root term from an equation, you can square both sides of the equation. This will get rid of the square root, but you need to be careful not to introduce any extraneous solutions.
Q: What is an extraneous solution?
A: An extraneous solution is a solution that is not valid for the original equation. When you square both sides of an equation, you may introduce extraneous solutions that are not valid for the original equation.
Q: How do I avoid extraneous solutions?
A: To avoid extraneous solutions, you need to check your solutions by plugging them back into the original equation. If the solution satisfies the original equation, then it is a valid solution.
Q: What are some common mistakes to avoid when solving equations involving square roots?
A: Some common mistakes to avoid when solving equations involving square roots include:
- Failing to isolate the square root term first
- Not squaring both sides of the equation
- Not adding or subtracting the same value from both sides of the equation
- Introducing extraneous solutions
Q: Can you give an example of an equation involving a square root that is not easily solvable?
A: Yes, here is an example of an equation involving a square root that is not easily solvable:
√(x^2 + 4x + 4) = 3
This equation involves a quadratic expression inside the square root, which makes it difficult to solve.
Q: How do I solve an equation involving a square root with a quadratic expression inside the square root?
A: To solve an equation involving a square root with a quadratic expression inside the square root, you can try the following:
- Factor the quadratic expression inside the square root
- Use the factored form to simplify the equation
- Solve for the variable using the simplified equation
Q: Can you give an example of an equation involving a square root with a quadratic expression inside the square root that is solvable?
A: Yes, here is an example of an equation involving a square root with a quadratic expression inside the square root that is solvable:
√(x^2 + 4x + 4) = 3
Factoring the quadratic expression inside the square root:
√((x + 2)^2) = 3
Simplifying the equation:
x + 2 = 3
Solving for x:
x = 1
Therefore, the solution to the equation is x = 1.
Conclusion
In this article, we answered some frequently asked questions related to solving equations involving square roots. We discussed the importance of isolating the square root term, squaring both sides of the equation, and avoiding extraneous solutions. We also provided examples of equations involving square roots that are not easily solvable and provided tips on how to solve them. By following these tips and techniques, you can become proficient in solving equations involving square roots.