Solve The Equation:$\sqrt{6x + 48} - 1 = 5$

Feb 28, 2025 by ADMIN 44 views

$Solving the Equation: $\sqrt{6x + 48} - 1 = 5$$

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving the equation $\sqrt{6x + 48} - 1 = 5$ . This equation involves a square root, and our goal is to isolate the variable $x$ and find its value.

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation $\sqrt{6x + 48} - 1 = 5$ states that the square root of $6x + 48$ minus 1 is equal to 5. Our goal is to find the value of $x$ that satisfies this equation.

Step 1: Isolate the Square Root

To solve the equation, we need to isolate the square root term. We can do this by adding 1 to both sides of the equation. This will give us:

$\sqrt{6x + 48} = 5 + 1$

$\sqrt{6x + 48} = 6$

Step 2: Square Both Sides

Now that we have isolated the square root term, we can square both sides of the equation to get rid of the square root. This will give us:

$6x + 48 = 36$

Step 3: Subtract 48 from Both Sides

Next, we need to subtract 48 from both sides of the equation to isolate the term with $x$ . This will give us:

$6x = 36 - 48$

$6x = -12$

Step 4: Divide Both Sides by 6

Finally, we need to divide both sides of the equation by 6 to solve for $x$ . This will give us:

$x = \frac{-12}{6}$

$x = -2$

Conclusion

In this article, we solved the equation $\sqrt{6x + 48} - 1 = 5$ by isolating the square root term, squaring both sides, subtracting 48 from both sides, and finally dividing both sides by 6. We found that the value of $x$ that satisfies this equation is $x = -2$ .

Tips and Tricks

When solving equations involving square roots, it's essential to remember the following tips and tricks:

Always isolate the square root term before squaring both sides.
When squaring both sides, make sure to square both the positive and negative terms.
When subtracting or adding terms, make sure to do so on both sides of the equation.
Finally, when dividing both sides, make sure to divide both the positive and negative terms.

Real-World Applications

Solving equations involving square roots has many real-world applications. For example, in physics, we often encounter equations involving square roots when dealing with problems involving motion and energy. In engineering, we use equations involving square roots to design and optimize systems. In finance, we use equations involving square roots to calculate interest rates and investment returns.

Common Mistakes

When solving equations involving square roots, there are several common mistakes to avoid:

Not isolating the square root term before squaring both sides.
Squaring only one side of the equation.
Not subtracting or adding terms on both sides of the equation.
Not dividing both sides by the correct value.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article, you can solve equations involving square roots and apply them to real-world problems. Remember to always isolate the square root term, square both sides, subtract or add terms on both sides, and finally divide both sides by the correct value. With practice and patience, you can become proficient in solving equations involving square roots.

Introduction

In our previous article, we solved the equation $\sqrt{6x + 48} - 1 = 5$ by isolating the square root term, squaring both sides, subtracting 48 from both sides, and finally dividing both sides by 6. We found that the value of $x$ that satisfies this equation is $x = -2$ . In this article, we will answer some frequently asked questions about solving equations involving square roots.

Q&A

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 adding or subtracting a constant from both sides of the equation.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to get rid of the square root. Squaring both sides allows us to eliminate the square root and solve for the variable.

Q: What is the difference between squaring a positive and negative term?

A: When squaring a positive term, the result is always positive. When squaring a negative term, the result is always positive. For example, $(3)^2 = 9$ and $(-3)^2 = 9$ .

Q: Why do we need to subtract or add terms on both sides of the equation?

A: We need to subtract or add terms on both sides of the equation to maintain the equality of the equation. If we subtract or add a term on only one side of the equation, the equation will no longer be true.

Q: What is the final step in solving an equation involving a square root?

A: The final step in solving an equation involving a square root is to divide both sides of the equation by the coefficient of the variable. This will give us the value of the variable.

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 not isolating the square root term, squaring only one side of the equation, not subtracting or adding terms on both sides of the equation, and not dividing both sides by the correct value.

Q: How do I know if I have solved the equation correctly?

A: To know if you have solved the equation correctly, you can plug the value of the variable back into the original equation and check if it is true. If the equation is true, then you have solved it correctly.