Solving The Equation √{8-7z} - √{z^2+20} = 0 A Step-by-Step Guide

In this article, we will delve into the intricacies of solving the equation √{8-7z} - √{z^2+20} = 0. This equation, which involves square roots and algebraic expressions, requires a systematic approach to arrive at the correct solutions. We will explore the necessary steps, potential pitfalls, and the underlying mathematical principles that govern the solution process. Understanding how to solve such equations is crucial for anyone studying algebra and beyond, as it enhances problem-solving skills and reinforces fundamental mathematical concepts. This comprehensive guide aims to equip you with the knowledge and techniques to tackle similar problems with confidence. Our main goal is to ensure that every step is crystal clear, making it easier for students and enthusiasts alike to grasp the solution.

Understanding the Equation

Before we jump into the solution, it's crucial to understand the type of equation we are dealing with. The equation √{8-7z} - √{z^2+20} = 0 is a radical equation, specifically one involving square roots. Radical equations require careful handling because squaring both sides, a common technique for solving them, can introduce extraneous solutions. These are solutions that satisfy the transformed equation but not the original one. Therefore, it's essential to check all potential solutions in the original equation. In this particular equation, we have two square root terms, one linear (8-7z) and one quadratic (z^2+20). The interplay between these terms is what makes the solution process interesting. We will break down each step, making sure to highlight the importance of verifying solutions at the end. Additionally, we must consider the domain of the square root functions. The expressions inside the square roots must be non-negative, meaning that 8-7z ≥ 0 and z^2+20 ≥ 0. This gives us a preliminary understanding of the possible values of z that we should consider. By addressing these initial considerations, we set ourselves up for a more organized and accurate solution process. The equation's structure tells us that isolating and squaring terms will be key, but we need to do so judiciously.

Step-by-Step Solution

Isolating the Square Roots

The first step in solving the equation √{8-7z} - √{z^2+20} = 0 is to isolate one of the square root terms. This makes it easier to eliminate the radicals when we square both sides. To do this, we can add √{z^2+20} to both sides of the equation, which gives us:

√{8-7z} = √{z^2+20}

This rearrangement sets the stage for the next crucial step: squaring both sides of the equation. Isolating the square roots ensures that when we square both sides, we eliminate the square roots directly without creating more complex terms.

Squaring Both Sides

Now that we have isolated the square roots, the next step is to eliminate them by squaring both sides of the equation. This is a standard technique for solving radical equations, but it's crucial to remember that it can introduce extraneous solutions. Squaring both sides of √{8-7z} = √{z^2+20} gives us:

(√{8-7z})^2 = (√{z^2+20})2

Simplifying this, we get:

8-7z = z^2+20

This transformation eliminates the square roots, turning the original equation into a quadratic equation, which we can solve using standard algebraic methods. This step is pivotal, as it converts a more complex radical equation into a familiar form.

Rearranging into a Quadratic Equation

After squaring both sides, we are left with 8-7z = z^2+20. To solve this, we need to rearrange it into a standard quadratic equation form, which is ax^2 + bx + c = 0. To do this, we can subtract 8 and add 7z to both sides of the equation:

0 = z^2 + 7z + 12

Now we have a quadratic equation in the standard form, which is much easier to solve. This rearrangement is a key step in simplifying the equation to a solvable format.

Solving the Quadratic Equation

The quadratic equation we have is z^2 + 7z + 12 = 0. There are several methods to solve a quadratic equation, including factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We are looking for two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4. Therefore, we can factor the quadratic equation as:

(z + 3)(z + 4) = 0

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

z + 3 = 0 or z + 4 = 0

Solving these linear equations, we get:

z = -3 or z = -4

These are the potential solutions to our original radical equation. However, we must check these solutions to ensure they are not extraneous.

Checking for Extraneous Solutions

As mentioned earlier, squaring both sides of an equation can introduce extraneous solutions. Therefore, it's crucial to check each potential solution in the original equation √{8-7z} - √{z^2+20} = 0. We will check each value of z separately.

Checking z = -3

Substitute z = -3 into the original equation:

√{8-7(-3)} - √{(-3)^2+20} = √{8+21} - √{9+20} = √{29} - √{29} = 0

Since the equation holds true, z = -3 is a valid solution.

Checking z = -4

Substitute z = -4 into the original equation:

√{8-7(-4)} - √{(-4)^2+20} = √{8+28} - √{16+20} = √{36} - √{36} = 6 - 6 = 0

Since the equation holds true, z = -4 is also a valid solution.

Final Solutions

After checking both potential solutions, we find that both z = -3 and z = -4 satisfy the original equation. Therefore, the solutions to the equation √{8-7z} - √{z^2+20} = 0 are:

• z = -3
• z = -4

This thorough check ensures that we have not included any extraneous solutions and that our answers are correct. The systematic approach, from isolating square roots to verifying solutions, is vital in solving radical equations accurately.

Conclusion

In conclusion, solving the equation √{8-7z} - √{z^2+20} = 0 involved several key steps, each vital for reaching the correct solutions. We began by isolating the square root terms, then squared both sides to eliminate the radicals, transforming the equation into a quadratic form. Rearranging the equation into the standard quadratic form allowed us to solve it by factoring. We found potential solutions z = -3 and z = -4, but the process didn't end there. The crucial step of checking for extraneous solutions confirmed that both values are indeed valid solutions. This meticulous approach highlights the importance of not only finding potential solutions but also verifying them in the original equation, a step that is paramount when dealing with radical equations. Understanding these steps and the underlying principles equips you with the tools to confidently tackle similar problems. The final solutions, z = -3 and z = -4, demonstrate the effectiveness of a systematic and thorough method in solving algebraic equations. By mastering these techniques, students and enthusiasts can enhance their problem-solving skills and deepen their understanding of mathematical concepts.

Final Answer

The solutions to the equation √{8-7z} - √{z^2+20} = 0 are:

• A. -4
• B. -3