Solve For { X $} : : : { \sqrt{5x + 56} - 3 = 1 \}

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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 $\sqrt{5x + 56} - 3 = 1$ for the variable $x$. This equation involves a square root, and our goal is to isolate the variable $x$ and find its value.

Understanding the Equation

The given equation is $\sqrt{5x + 56} - 3 = 1$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The first step is to add $3$ to both sides of the equation to get rid of the negative term.

Adding 3 to Both Sides

$\sqrt{5x + 56} - 3 + 3 = 1 + 3$

This simplifies to:

$\sqrt{5x + 56} = 4$

Removing the Square Root

The next step is to remove the square root from the equation. To do this, we need to square both sides of the equation.

Squaring Both Sides

$(\sqrt{5x + 56})^2 = 4^2$

This simplifies to:

$5x + 56 = 16$

Solving for $x$

Now that we have the equation $5x + 56 = 16$, we can solve for $x$. The first step is to subtract $56$ from both sides of the equation.

Subtracting 56 from Both Sides

$5x + 56 - 56 = 16 - 56$

This simplifies to:

$5x = -40$

Isolating $x$

The final step is to isolate $x$ by dividing both sides of the equation by $5$.

Dividing Both Sides by 5

$\frac{5x}{5} = \frac{-40}{5}$

This simplifies to:

$x = -8$

Conclusion

In this article, we solved the equation $\sqrt{5x + 56} - 3 = 1$ for the variable $x$. We started by adding $3$ to both sides of the equation to get rid of the negative term, then squared both sides to remove the square root. Finally, we solved for $x$ by subtracting $56$ from both sides and dividing both sides by $5$. The value of $x$ is $-8$.

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 remove the square root, but it can also introduce extraneous solutions.
• Be careful when subtracting or adding terms to both sides of the equation, as this can affect the value of the equation.

Common Mistakes

• Failing to isolate the square root term before squaring both sides of the equation.
• Introducing extraneous solutions by squaring both sides of the equation.
• Not checking for extraneous solutions after solving for $x$.

Real-World Applications

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

• Physics: Solving equations involving square roots can help model real-world phenomena, such as the motion of objects under the influence of gravity.
• Engineering: Solving equations involving square roots can help design and optimize systems, such as bridges and buildings.
• Computer Science: Solving equations involving square roots can help develop algorithms and data structures, such as those used in computer graphics and game development.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. By following the steps outlined in this article, you can solve equations involving square roots and develop a deeper understanding of algebraic equations. Remember to be careful when subtracting or adding terms to both sides of the equation, and always check for extraneous solutions after solving for $x$.

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Introduction

In our previous article, we solved the equation $\sqrt{5x + 56} - 3 = 1$ for the variable $x$. We received many questions from readers who were struggling to understand the solution. In this article, we will address some of the most frequently asked questions and provide additional explanations to help clarify the solution.

Q&A

Q: What is the first step in solving the equation $\sqrt{5x + 56} - 3 = 1$?

A: The first step is to add $3$ to both sides of the equation to get rid of the negative term. This gives us $\sqrt{5x + 56} = 4$.

Q: Why do we need to add $3$ to both sides of the equation?

A: We need to add $3$ to both sides of the equation to isolate the square root term. By doing so, we can remove the negative term and simplify the equation.

Q: What happens if we don't add $3$ to both sides of the equation?

A: If we don't add $3$ to both sides of the equation, we will be left with a negative term, which can make it difficult to solve the equation.

Q: How do we remove the square root from the equation?

A: We remove the square root from the equation by squaring both sides of the equation. This gives us $5x + 56 = 16$.

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

A: We need to square both sides of the equation to remove the square root. By doing so, we can simplify the equation and solve for $x$.

Q: What happens if we don't square both sides of the equation?

A: If we don't square both sides of the equation, we will be left with a square root term, which can make it difficult to solve the equation.

Q: How do we solve for $x$?

A: We solve for $x$ by subtracting $56$ from both sides of the equation and then dividing both sides by $5$. This gives us $x = -8$.

Q: Why do we need to subtract $56$ from both sides of the equation?

A: We need to subtract $56$ from both sides of the equation to isolate the term with $x$. By doing so, we can solve for $x$.

Q: Why do we need to divide both sides of the equation by $5$?

A: We need to divide both sides of the equation by $5$ to solve for $x$. By doing so, we can find the value of $x$.

Q: What is the final answer?

A: The final answer is $x = -8$.

Additional 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 remove the square root, but it can also introduce extraneous solutions.
• Be careful when subtracting or adding terms to both sides of the equation, as this can affect the value of the equation.
• Always check for extraneous solutions after solving for $x$.

Common Mistakes

• Failing to isolate the square root term before squaring both sides of the equation.
• Introducing extraneous solutions by squaring both sides of the equation.
• Not checking for extraneous solutions after solving for $x$.

Real-World Applications

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

• Physics: Solving equations involving square roots can help model real-world phenomena, such as the motion of objects under the influence of gravity.
• Engineering: Solving equations involving square roots can help design and optimize systems, such as bridges and buildings.
• Computer Science: Solving equations involving square roots can help develop algorithms and data structures, such as those used in computer graphics and game development.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. By following the steps outlined in this article, you can solve equations involving square roots and develop a deeper understanding of algebraic equations. Remember to be careful when subtracting or adding terms to both sides of the equation, and always check for extraneous solutions after solving for $x$.