Solving 3.5√(1.7x + 7) + 2.5 = 14.4 A Step By Step Guide

by THE IDEN 57 views

Introduction

In this article, we will delve into the process of solving the equation 3.5√(1.7x + 7) + 2.5 = 14.4. This equation involves a square root, which requires a systematic approach to isolate the variable x and find its value. We will break down the steps, explain the underlying principles, and provide a detailed solution. Additionally, we will emphasize the importance of checking our solution to ensure it satisfies the original equation. This comprehensive guide is designed to help students, educators, and anyone interested in mathematics to understand and solve such equations effectively.

Understanding the Equation

Before we dive into the solution, let's understand the equation. The equation 3.5√(1.7x + 7) + 2.5 = 14.4 is a radical equation, specifically one involving a square root. Radical equations require us to isolate the radical term and then square both sides to eliminate the square root. It's crucial to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, checking our solution at the end is a vital step.

The equation consists of several components. We have a variable, x, which we aim to find. The expression 1.7x + 7 is under the square root, indicating that we need to isolate this term before we can deal with the square root. The square root term is multiplied by 3.5, and 2.5 is added to the result. The entire expression is set equal to 14.4. Our goal is to reverse these operations step by step to isolate x.

Step-by-Step Solution

Let's begin with the step-by-step solution to the equation 3.5√(1.7x + 7) + 2.5 = 14.4.

Step 1: Isolate the Radical Term

Our first goal is to isolate the term containing the square root. To do this, we need to subtract 2.5 from both sides of the equation. This will remove the constant term on the left side and bring us closer to isolating the square root.

3.5√(1.7x + 7) + 2.5 - 2.5 = 14.4 - 2.5

This simplifies to:

3. 5√(1.7x + 7) = 11.9

Step 2: Divide by the Coefficient

Next, we need to isolate the square root itself. To do this, we divide both sides of the equation by 3.5, which is the coefficient multiplying the square root term.

(3.5√(1.7x + 7)) / 3.5 = 11.9 / 3.5

This simplifies to:

√(1.7x + 7) = 3.4

Step 3: Square Both Sides

Now that we have isolated the square root, we can eliminate it by squaring both sides of the equation. This will remove the square root and allow us to work with the expression inside it.

(√(1.7x + 7))² = (3.4)²

This simplifies to:

1.7x + 7 = 11.56

Step 4: Isolate the Variable Term

We now have a linear equation. To isolate the term with x, we subtract 7 from both sides of the equation.

1.7x + 7 - 7 = 11.56 - 7

This simplifies to:

1.7x = 4.56

Step 5: Solve for x

Finally, to solve for x, we divide both sides of the equation by 1.7.

(1.7x) / 1.7 = 4.56 / 1.7

This simplifies to:

x = 2.68235...

Rounding to a reasonable number of decimal places, we get:

x ≈ 2.68

Checking the Solution

As mentioned earlier, it's crucial to check our solution to ensure it satisfies the original equation. Substitute the value of x we found back into the original equation and see if both sides are equal.

Original equation:

3. 5√(1.7x + 7) + 2.5 = 14.4

Substitute x ≈ 2.68:

3. 5√(1.7(2.68) + 7) + 2.5

First, calculate inside the parenthesis:

1. 7(2.68) ≈ 4.556

Add 7:

4. 556 + 7 = 11.556

Now, take the square root:

√(11.556) ≈ 3.4

Multiply by 3.5:

3. 5(3.4) = 11.9

Finally, add 2.5:

11. 9 + 2.5 = 14.4

The left side of the equation equals 14.4, which is the same as the right side. Therefore, our solution x ≈ 2.68 is correct.

Common Mistakes and How to Avoid Them

When solving radical equations, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution.

Mistake 1: Forgetting to Isolate the Radical Term

One of the most common mistakes is squaring both sides of the equation before isolating the radical term. This can lead to a more complex equation that is harder to solve. Always isolate the radical term first before squaring.

Mistake 2: Incorrectly Squaring Both Sides

When squaring both sides of the equation, make sure to square the entire side, not just individual terms. For example, if you have (a + b)², it's not equal to a² + b². It should be a² + 2ab + b². This is a crucial algebraic rule to remember.

Mistake 3: Not Checking for Extraneous Solutions

As mentioned earlier, squaring both sides of an equation can introduce extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. Always check your solution by substituting it back into the original equation.

Mistake 4: Arithmetic Errors

Simple arithmetic errors can derail your solution. Be careful with your calculations, especially when dealing with decimals and square roots. Double-check your work to minimize the chances of making a mistake.

Tips for Solving Radical Equations

Here are some helpful tips to keep in mind when solving radical equations:

  1. Isolate the Radical: Always start by isolating the radical term on one side of the equation.
  2. Square Both Sides: After isolating the radical, square both sides of the equation to eliminate the square root.
  3. Solve the Resulting Equation: Solve the equation that results after squaring both sides. This might be a linear equation, a quadratic equation, or another type of equation.
  4. Check Your Solution: Always check your solution in the original equation to make sure it is not an extraneous solution.
  5. Be Careful with Signs: Pay close attention to the signs of the terms in the equation, especially when squaring both sides.
  6. Double-Check Your Work: Review your steps and calculations to minimize the chances of making errors.

Real-World Applications

Radical equations are not just abstract mathematical concepts; they have practical applications in various fields. Understanding how to solve them can be useful in real-world scenarios.

Physics

In physics, radical equations are used in problems involving velocity, acceleration, and distance. For example, the velocity of an object in free fall can be calculated using a formula that involves a square root.

Engineering

Engineers use radical equations in structural design, fluid dynamics, and electrical engineering. For example, the period of a pendulum can be calculated using a formula that involves a square root.

Finance

In finance, radical equations can be used to calculate rates of return and compound interest. Understanding these concepts is crucial for making informed financial decisions.

Computer Graphics

Radical equations are used in computer graphics for rendering and transformations. They help in calculating distances and creating realistic visual effects.

Conclusion

In summary, solving the equation 3.5√(1.7x + 7) + 2.5 = 14.4 involves a systematic approach of isolating the radical term, squaring both sides, and solving the resulting equation. It’s vital to check the solution to avoid extraneous solutions. By following the steps outlined in this guide and being mindful of common mistakes, you can confidently solve radical equations. Understanding these equations not only strengthens your mathematical skills but also prepares you for real-world applications in various fields. Practice is key to mastering these techniques, so keep solving equations and challenging yourself.