Solving The Equation √(-3v + 30) = V - 4 A Step-by-Step Guide

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In this comprehensive article, we will delve into the step-by-step process of solving the equation √(-3v + 30) = v - 4 for real number solutions. This equation, which involves a square root, requires careful algebraic manipulation and verification to ensure we arrive at the correct solution(s). We will thoroughly explore the necessary steps, check for extraneous solutions, and provide a detailed explanation to enhance your understanding of solving such equations. Whether you're a student tackling algebra problems or simply someone keen to brush up on your mathematical skills, this guide will provide you with the clarity and confidence to solve similar problems effectively.

Step 1: Isolate the Square Root

The given equation is √(-3v + 30) = v - 4. Our first goal is to isolate the square root term on one side of the equation. In this case, the square root is already isolated, which simplifies our initial step.

Step 2: Square Both Sides

To eliminate the square root, we square both sides of the equation. This operation allows us to transition from an equation involving a square root to a more manageable algebraic form. Squaring both sides gives us:

(√(-3v + 30))² = (v - 4)²

Simplifying this, we get:

-3v + 30 = (v - 4)²

Step 3: Expand and Simplify

Next, we need to expand the right side of the equation and simplify. The term (v - 4)² can be expanded as follows:

(v - 4)² = v² - 8v + 16

Now, substitute this back into our equation:

-3v + 30 = v² - 8v + 16

To solve for v, we need to set the equation to zero by moving all terms to one side. We can do this by adding 3v and subtracting 30 from both sides:

0 = v² - 8v + 16 + 3v - 30

Combine like terms to simplify:

0 = v² - 5v - 14

Step 4: Solve the Quadratic Equation

We now have a quadratic equation in the form of v² - 5v - 14 = 0. To solve this, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is the most straightforward approach.

We look for two numbers that multiply to -14 and add to -5. These numbers are -7 and 2. Thus, we can factor the quadratic equation as:

(v - 7)(v + 2) = 0

Setting each factor equal to zero gives us the potential solutions:

  • v - 7 = 0 => v = 7
  • v + 2 = 0 => v = -2

Step 5: Check for Extraneous Solutions

It is crucial to check our solutions in the original equation to ensure they are valid. Squaring both sides of an equation can introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one.

Check v = 7

Substitute v = 7 into the original equation √(-3v + 30) = v - 4:

√(-3(7) + 30) = 7 - 4

√(-21 + 30) = 3

√9 = 3

3 = 3

This solution is valid.

Check v = -2

Substitute v = -2 into the original equation √(-3v + 30) = v - 4:

√(-3(-2) + 30) = -2 - 4

√(6 + 30) = -6

√36 = -6

6 = -6

This solution is extraneous because 6 does not equal -6.

Step 6: Final Solution

After checking both potential solutions, we find that v = 7 is the only valid solution. The extraneous solution v = -2 must be discarded.

In summary, solving equations involving square roots requires careful attention to detail. The steps include isolating the square root, squaring both sides, simplifying the resulting equation, solving for the variable, and, most importantly, checking for extraneous solutions. By following these steps methodically, we can confidently determine the correct solution(s). In this case, the only real solution for the equation √(-3v + 30) = v - 4 is v = 7.

Solving equations with square roots can be tricky, but with a systematic approach and careful attention to detail, you can master them. Here are some additional tips to help you along the way:

  • Always Isolate the Square Root: Before squaring both sides, make sure the square root term is isolated on one side of the equation. This simplifies the process and reduces the chances of making errors.
  • Check for Extraneous Solutions: Squaring both sides of an equation can introduce extraneous solutions. Always substitute your potential solutions back into the original equation to verify their validity.
  • Simplify Before Squaring: If possible, simplify the equation before squaring both sides. This can make the algebra easier and reduce the complexity of the resulting equation.
  • Be Careful with Signs: Pay close attention to signs, especially when dealing with negative numbers and subtraction. A small error in sign can lead to an incorrect solution.
  • Use Factoring or the Quadratic Formula: When solving quadratic equations, factoring is often the quickest method. However, if factoring is not straightforward, use the quadratic formula or complete the square.
  • Practice Regularly: Like any mathematical skill, solving equations with square roots becomes easier with practice. Work through a variety of examples to build your confidence and proficiency.
  • Understand the Domain: Keep in mind that the expression inside the square root must be non-negative. This can help you identify potential solutions that are not valid.
  • Review Your Work: Always review your steps to catch any errors. This is particularly important in multi-step problems.

By following these tips and practicing regularly, you'll become more comfortable and skilled at solving equations involving square roots.

When solving equations with square roots, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Forgetting to Isolate the Square Root: Failing to isolate the square root before squaring both sides can lead to more complex equations and increase the likelihood of errors.
  • Not Checking for Extraneous Solutions: This is perhaps the most common mistake. Always check your solutions in the original equation to ensure they are valid.
  • Squaring Terms Incorrectly: When squaring a binomial, remember to use the correct formula (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b². Avoid the common mistake of simply squaring each term individually.
  • Sign Errors: Pay close attention to signs throughout the problem. A small sign error can lead to an incorrect solution.
  • Incorrectly Factoring Quadratic Equations: If you choose to solve a quadratic equation by factoring, make sure you factor it correctly. Double-check your factors to ensure they multiply to the correct quadratic expression.
  • Not Simplifying: Simplifying the equation before performing operations can make the problem easier to solve. Look for opportunities to combine like terms or reduce fractions.
  • Dividing by a Variable: Avoid dividing both sides of an equation by a variable unless you know the variable is not zero. Dividing by a variable can lead to the loss of solutions.
  • Misunderstanding the Domain: Remember that the expression inside a square root must be non-negative. Ignoring this can lead to invalid solutions.

By being aware of these common mistakes, you can take steps to avoid them and improve your accuracy in solving equations with square roots.

Square root equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields. Understanding these applications can provide a deeper appreciation for the importance of this topic.

  • Physics: Square root equations are used extensively in physics. For example, the period of a simple pendulum is given by the formula T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. Solving for L or g involves working with a square root equation.
  • Engineering: Engineers use square root equations in various calculations, such as determining the velocity of fluid flow through an orifice or the stress in a structural member.
  • Finance: In finance, the Black-Scholes model for option pricing involves square root equations. This model is used to estimate the fair price of European-style options.
  • Computer Graphics: Square root equations are used in computer graphics for calculating distances and creating realistic visual effects.
  • Navigation: The distance to the horizon can be approximated using a square root equation. This is useful in navigation and surveying.
  • Astronomy: Astronomers use square root equations to calculate orbital speeds and distances in space.
  • Statistics: The standard deviation, a measure of the spread of data, is calculated using a square root equation.

By recognizing these real-world applications, you can see that square root equations are not just theoretical exercises but valuable tools for solving practical problems in a variety of disciplines. This understanding can motivate you to master the techniques for solving these equations and appreciate their significance.

To solidify your understanding of solving equations with square roots, here are some practice problems. Work through these problems, applying the steps and tips discussed in this article. Be sure to check your solutions for extraneous roots.

  1. √(2x + 5) = x - 5
  2. √(3x - 2) = x - 2
  3. √(4x + 1) = x - 1
  4. √(5x - 4) = x - 4
  5. √(6x + 7) = x - 7
  6. √(x + 10) = x + 4
  7. √(2x + 15) = x + 6
  8. √(3x + 10) = x + 4
  9. √(4x - 4) = x - 1
  10. √(5x - 9) = x - 3

For each problem, follow these steps:

  • Isolate the square root.
  • Square both sides of the equation.
  • Simplify and solve the resulting equation.
  • Check your solutions in the original equation.
  • Identify and discard any extraneous solutions.

By working through these practice problems, you'll reinforce your understanding of the process and develop your problem-solving skills. Remember to pay attention to detail and check your work carefully.

Mastering the art of solving equations with square roots is a valuable skill in mathematics. By understanding the steps involved, being mindful of potential pitfalls, and practicing regularly, you can confidently tackle these problems. Remember to always isolate the square root, square both sides, simplify, solve, and check for extraneous solutions. With dedication and practice, you'll become proficient in solving these types of equations and appreciate their wide-ranging applications in various fields. Keep practicing, and you'll find that solving equations with square roots becomes second nature!