Finding Zeros Of 3√3x² - 19x + 10√3 A Step By Step Guide

In this comprehensive guide, we will delve into the process of finding the zeros of the quadratic equation 3√3x² - 19x + 10√3. Quadratic equations, characterized by their highest power of 2, are fundamental in mathematics and have wide-ranging applications in various fields, including physics, engineering, and computer science. The zeros of a quadratic equation, also known as its roots, are the values of the variable (in this case, 'x') that make the equation equal to zero. These zeros represent the points where the parabola represented by the quadratic equation intersects the x-axis. Mastering the techniques for finding zeros is crucial for solving real-world problems and gaining a deeper understanding of mathematical concepts. We will explore the steps involved in solving this specific equation, providing clear explanations and examples to ensure a thorough understanding of the process.

Understanding Quadratic Equations and Zeros

Quadratic equations are polynomial equations of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. The term 'zeros' refers to the values of 'x' that satisfy the equation, making the entire expression equal to zero. These zeros are also known as roots or solutions of the quadratic equation. Graphically, the zeros represent the x-intercepts of the parabola defined by the quadratic equation. The number of real zeros a quadratic equation can have is either two, one (a repeated root), or none, depending on the discriminant (b² - 4ac). When the discriminant is positive, there are two distinct real roots; when it is zero, there is one repeated real root; and when it is negative, there are no real roots, but there are two complex roots. Understanding the relationship between the discriminant and the nature of the roots is crucial in solving quadratic equations efficiently. In the context of real-world applications, zeros can represent various physical quantities, such as the time at which a projectile hits the ground or the equilibrium points in a system.

To further clarify the concept, let's consider a simple quadratic equation like x² - 5x + 6 = 0. The zeros of this equation are the values of x that make the equation true. By factoring the quadratic expression, we get (x - 2)(x - 3) = 0. Setting each factor equal to zero gives us x - 2 = 0 and x - 3 = 0, which yield the solutions x = 2 and x = 3. These are the zeros of the equation, and they represent the points where the parabola y = x² - 5x + 6 intersects the x-axis. Similarly, for the equation we are going to solve, 3√3x² - 19x + 10√3 = 0, our goal is to find the values of x that satisfy this equation. This involves using methods such as factoring, completing the square, or the quadratic formula, each of which provides a systematic approach to finding the zeros. The complexity of the coefficients in our equation (including the square root terms) may make factoring more challenging, but with a careful step-by-step approach, we can determine the zeros accurately.

Step 1: Factoring the Quadratic Equation

The first step in finding the zeros of the quadratic equation 3√3x² - 19x + 10√3 is to attempt factoring. Factoring involves expressing the quadratic equation as a product of two binomials. This method is often the most efficient when the coefficients of the quadratic equation are integers or simple fractions, but it can become more complex when dealing with irrational numbers, such as √3 in our case. The key to successful factoring is to find two numbers that multiply to give the product of the leading coefficient (3√3) and the constant term (10√3), and that add up to the middle coefficient (-19). This can be a bit like solving a puzzle, requiring careful consideration of the factors and their combinations. In this particular equation, the presence of √3 complicates the factoring process, but by systematically exploring the possibilities, we can identify the correct factors. This initial step is critical, as it sets the stage for finding the zeros of the equation and provides a direct path to the solutions if the factoring is done correctly.

To factor the given equation 3√3x² - 19x + 10√3, we need to find two numbers that multiply to (3√3)(10√3) = 90 and add up to -19. Let's consider the factors of 90: 1 and 90, 2 and 45, 3 and 30, 5 and 18, 6 and 15, 9 and 10. Among these pairs, 9 and 10 seem promising since their sum is close to 19. We need a sum of -19, so we consider -9 and -10. Indeed, (-9) * (-10) = 90 and (-9) + (-10) = -19. Now we rewrite the middle term -19x using these numbers: 3√3x² - 9x - 10x + 10√3. Next, we factor by grouping. From the first two terms, we can factor out 3x, and from the last two terms, we can factor out -10√3. This gives us 3x(√3x - 3) - 10(x - √3). To proceed further, we need to manipulate the terms inside the parentheses to find a common factor. Notice that 3 can be written as √3 * √3, so √3x - 3 can be rewritten as √3x - √3 * √3, which equals √3(x - √3). Now our equation looks like this: 3√3x² - 9x - 10x + 10√3 = 3x(√3x - 3) - 10(x - √3) = 3x√3(x/√3 - 1) - 10(x - √3). This is not directly leading to a simple factorization, so we made a mistake earlier. Let's revisit the factorization by grouping step and adjust our approach to correctly factor the equation.

Step 2: Adjusting the Factoring Approach

After our initial attempt at factoring revealed a complication, we need to refine our approach. The key is to correctly split the middle term (-19x) using the two numbers we identified (-9 and -10) in such a way that we can factor by grouping effectively. The original split led to terms that didn't allow for a clean factorization, so we'll revisit the grouping strategy. Instead of directly substituting -9x and -10x, we should look for coefficients that include √3 to match the terms in the original equation. This will allow us to factor out common factors more easily. The goal remains the same: to express the quadratic equation as a product of two binomials, but we'll need to be more strategic in how we manipulate the terms. This step is crucial because a well-executed factoring will lead directly to the zeros of the equation, simplifying the overall solution process. The adjustment in our approach highlights the importance of flexibility and careful observation in solving mathematical problems, particularly when dealing with equations involving irrational numbers.

To correctly factor 3√3x² - 19x + 10√3, we reconsider the split of the middle term. Instead of -9x and -10x, we look for terms that will allow us to factor out factors involving √3. We keep the numbers -10 and -9 in mind, but adjust their association with 'x' to create common factors. We can rewrite -19x as -10x - 9x. However, this didn't work in the previous attempt. So, let's consider expressing -19x as a sum involving terms with √3. We need to rewrite -19x as a combination that allows us to factor out common terms involving √3. Let's try rewriting the equation as 3√3x² - 10√3x + 1√3x - 19x + 10√3 and see if we can split -19x in a way that is conducive to factoring. This approach doesn't seem to simplify the factoring process. Let's reconsider the correct split of -19x using the numbers -10 and -9. We rewrite the middle term as -9x - 10x. Now we have 3√3x² - 9x - 10x + 10√3. We can rewrite -9x as -3√3 * √3x and 10√3 as 10 * √3. So, the equation becomes 3√3x² - 3√3(√3x) - 10x + 10√3. Now factor by grouping: 3√3x(x - √3) - 10(x - √3). We have a common factor of (x - √3). So, we can factor it out: (3√3x - 10)(x - √3). This is the correct factored form of the quadratic equation.

Step 3: Setting Factors to Zero and Solving for x

With the quadratic equation successfully factored into (3√3x - 10)(x - √3) = 0, the next step is to find the zeros. The zeros of the equation are the values of 'x' that make the equation true, which means we need to find the values of 'x' that make each factor equal to zero. This principle is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. By setting each factor equal to zero, we create two simpler equations that can be solved independently. This step is a direct application of a fundamental algebraic principle and allows us to transition from a complex quadratic equation to two linear equations, which are much easier to solve. Solving these linear equations will give us the values of 'x' that are the zeros (or roots) of the original quadratic equation. This step is critical in the solution process, as it transforms the problem from finding the zeros of a quadratic to solving basic linear equations.

To find the zeros, we set each factor equal to zero: 3√3x - 10 = 0 and x - √3 = 0. Solving the first equation, 3√3x - 10 = 0, we first isolate the term with 'x': 3√3x = 10. Then, we divide both sides by 3√3 to solve for x: x = 10 / (3√3). To rationalize the denominator, we multiply the numerator and the denominator by √3: x = (10√3) / (3√3 * √3) = (10√3) / (3 * 3) = (10√3) / 9. So, the first zero is x = (10√3) / 9. Now, let's solve the second equation, x - √3 = 0. Adding √3 to both sides gives us x = √3. So, the second zero is x = √3. Therefore, the zeros of the quadratic equation 3√3x² - 19x + 10√3 are x = (10√3) / 9 and x = √3. These are the values of 'x' that make the original equation equal to zero.

Step 4: Verifying the Zeros

After finding the zeros of the quadratic equation, it is crucial to verify these solutions to ensure accuracy. Verification involves substituting each zero back into the original equation and confirming that the equation holds true (i.e., equals zero). This step is essential because it can reveal any errors made during the factoring or solving process. Errors can occur due to incorrect algebraic manipulations, sign mistakes, or miscalculations. By verifying the zeros, we gain confidence in the correctness of our solutions and ensure that we have accurately solved the quadratic equation. This process also reinforces our understanding of the relationship between the zeros and the equation itself. Verification is not merely a procedural step; it is an integral part of problem-solving in mathematics, providing a check on our work and enhancing our comprehension of the concepts involved.

To verify the zeros, we substitute each value of x back into the original equation 3√3x² - 19x + 10√3 = 0. First, let's verify x = (10√3) / 9: 3√3 * ((10√3) / 9)² - 19 * ((10√3) / 9) + 10√3 = 3√3 * (100 * 3) / 81 - (190√3) / 9 + 10√3 = (300√3) / 81 - (190√3) / 9 + 10√3. To simplify, we find a common denominator, which is 81: (300√3) / 81 - (190√3 * 9) / 81 + (10√3 * 81) / 81 = (300√3 - 1710√3 + 810√3) / 81 = (-600√3) / 81. There seems to be a mistake in the calculation. Let's correct the fractions: (300√3)/81 - (190√3)/9 + 10√3 = (100√3)/27 - (190√3)/9 + 10√3 = (100√3)/27 - (570√3)/27 + (270√3)/27 = (100√3 - 570√3 + 270√3) / 27 = (-200√3)/27. This still doesn't equal 0, which indicates an error either in the simplification or in the zero itself. Let's verify x = √3: 3√3 * (√3)² - 19 * √3 + 10√3 = 3√3 * 3 - 19√3 + 10√3 = 9√3 - 19√3 + 10√3 = (9 - 19 + 10)√3 = 0. So, x = √3 is a correct zero. We need to re-evaluate our calculations for x = (10√3) / 9. Upon reviewing, the mistake was in the calculation of (300√3) / 81, which simplifies to (100√3) / 27. The correct calculation is: (100√3)/27 - (190√3)/9 + 10√3 = (100√3)/27 - (570√3)/27 + (270√3)/27 = (100√3 - 570√3 + 270√3)/27 = (-200√3)/27. This result is still incorrect. The mistake is in assuming that (10√3)/9 is a root. Let's redo the verification for x = (10√3) / 9: 3√3 * ((10√3)/9)² - 19 * (10√3)/9 + 10√3 = 3√3 * (300/81) - (190√3)/9 + 10√3 = (900√3)/81 - (190√3)/9 + 10√3 = (100√3)/9 - (190√3)/9 + (90√3)/9 = (100√3 - 190√3 + 90√3)/9 = 0/9 = 0. So, x = (10√3) / 9 is also a correct zero.

Conclusion

In conclusion, finding the zeros of the quadratic equation 3√3x² - 19x + 10√3 involves a systematic approach that combines factoring techniques and the zero-product property. We successfully factored the equation into (3√3x - 10)(x - √3) and then set each factor to zero to solve for 'x'. This process yielded two zeros: x = (10√3) / 9 and x = √3. We then verified these zeros by substituting them back into the original equation, confirming that both values satisfy the equation. This step-by-step guide illustrates the importance of breaking down a complex problem into smaller, manageable steps and carefully executing each step to arrive at the correct solution. The techniques demonstrated here are applicable to a wide range of quadratic equations, making this a valuable skill for anyone studying mathematics or related fields. The verification step further underscores the need for precision and accuracy in mathematical problem-solving, ensuring the reliability of the results obtained.

This detailed exploration not only provides a solution to the specific equation but also enhances understanding of quadratic equations and their solutions in general. The combination of factoring, solving linear equations, and verification forms a robust approach to tackling such problems. Moreover, the process highlights the significance of checking one's work to avoid errors and build confidence in the solutions. The ability to find zeros of quadratic equations is fundamental in various applications, from physics and engineering to economics and computer science, making this a crucial mathematical skill to master.