Solve For \[$ X \$\]:$\[ \sqrt{3x + 76} + 7 = 15 \\]

$$$$

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will focus on solving the equation $\sqrt{3x + 76} + 7 = 15$ to find the value of $x$. This equation involves a square root, and our goal is to isolate the variable $x$.

Step 1: Isolate the Square Root

The first step in solving this equation is to isolate the square root term. To do this, we need to subtract 7 from both sides of the equation. This will give us:

$\sqrt{3x + 76} = 15 - 7$

$\sqrt{3x + 76} = 8$

Step 2: Square Both Sides

Now that we have isolated the square root term, we can square both sides of the equation to eliminate the square root. Squaring both sides gives us:

$(\sqrt{3x + 76})^2 = 8^2$

$3x + 76 = 64$

Step 3: Subtract 76 from Both Sides

Next, we need to subtract 76 from both sides of the equation to isolate the term with the variable $x$. This gives us:

$3x = 64 - 76$

$3x = -12$

Step 4: Divide Both Sides by 3

Finally, we need to divide both sides of the equation by 3 to solve for $x$. This gives us:

$x = \frac{-12}{3}$

$x = -4$

Conclusion

In this article, we solved the equation $\sqrt{3x + 76} + 7 = 15$ to find the value of $x$. We started by isolating the square root term, then squared both sides to eliminate the square root. We then subtracted 76 from both sides and finally divided both sides by 3 to solve for $x$. The final answer is $x = -4$.

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, but be careful not to introduce extraneous solutions.
• Always check your solutions by plugging them back into the original equation.

Real-World Applications

Solving equations involving square roots has many real-world applications, such as:

• Physics: When calculating the distance traveled by an object under the influence of gravity, square roots are often involved.
• Engineering: In designing structures, such as bridges or buildings, square roots are used to calculate stresses and loads.
• Finance: In calculating interest rates and investment returns, square roots are often used.

Common Mistakes

• Failing to isolate the square root term before squaring both sides.
• Introducing extraneous solutions by squaring both sides without checking the solutions.
• Not checking the solutions by plugging them back into the original equation.

Final Thoughts

Solving equations involving square roots can be challenging, but with the right approach, it can be done efficiently. By following the steps outlined in this article, you can solve equations like $\sqrt{3x + 76} + 7 = 15$ and find the value of $x$. Remember to always check your solutions and be careful not to introduce extraneous solutions.

$$$$

Introduction

In our previous article, we solved the equation $\sqrt{3x + 76} + 7 = 15$ to find the value of $x$. 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 a constant to 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 eliminate the square root. This is because squaring both sides of an equation is a way to get rid of the square root symbol.

Q: What is the difference between squaring both sides of an equation and multiplying both sides by a constant?

A: Squaring both sides of an equation is different from multiplying both sides by a constant. When we square both sides of an equation, we are essentially getting rid of the square root symbol. When we multiply both sides by a constant, we are changing the value of the equation.

Q: How do I know if I have introduced an extraneous solution?

A: To check if you have introduced an extraneous solution, you need to plug the solution back into the original equation. If the solution does not satisfy the original equation, then it is an extraneous solution.

Q: What is the importance of checking solutions?

A: Checking solutions is crucial in solving equations involving square roots. If you do not check your solutions, you may end up with an extraneous solution, which can lead to incorrect conclusions.

Q: Can I use a calculator to solve equations involving square roots?

A: Yes, you can use a calculator to solve equations involving square roots. However, it is essential to check your solutions by plugging them back into the original equation to ensure that they are correct.

Q: How do I know if an equation involves a square root?

A: An equation involves a square root if it contains the symbol $\sqrt{}$. This symbol indicates that the expression inside the square root is being raised to the power of 1/2.

Q: Can I use the same steps to solve equations involving cube roots?

A: No, you cannot use the same steps to solve equations involving cube roots. Cube roots are different from square roots, and you need to use different techniques to solve equations involving cube roots.

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 before squaring both sides.
• Introducing extraneous solutions by squaring both sides without checking the solutions.
• Not checking the solutions by plugging them back into the original equation.

Q: How do I know if an equation is a quadratic equation or an equation involving a square root?

A: To determine if an equation is a quadratic equation or an equation involving a square root, you need to look at the terms in the equation. If the equation contains a term with a variable squared, it is a quadratic equation. If the equation contains a term with a variable inside a square root, it is an equation involving a square root.

Q: Can I use the quadratic formula to solve equations involving square roots?

A: No, you cannot use the quadratic formula to solve equations involving square roots. The quadratic formula is used to solve quadratic equations, not equations involving square roots.

Conclusion

Solving equations involving square roots can be challenging, but with the right approach, it can be done efficiently. By following the steps outlined in this article and avoiding common mistakes, you can solve equations like $\sqrt{3x + 76} + 7 = 15$ and find the value of $x$. Remember to always check your solutions and be careful not to introduce extraneous solutions.