Solving The Quadratic Equation 3y^2 + 5y - 8 = 0

Unveiling the Solutions to the Quadratic Equation: $3y^2 + 5y - 8 = 0$

In the realm of mathematics, quadratic equations stand as fundamental expressions, weaving their way through various disciplines. The equation at hand, $3y^2 + 5y - 8 = 0$, is a quintessential example, beckoning us to unravel its hidden solutions. Solving quadratic equations like this involves finding the values of the variable, in this case, 'y', that satisfy the equation, making the expression equal to zero. These values are also known as the roots or zeros of the quadratic equation.

There are several avenues we can explore to solve this equation, each with its unique charm and approach. Among the most prominent methods are factoring, completing the square, and the quadratic formula. Each method offers a different perspective on the equation, and the choice of method often depends on the specific characteristics of the equation and the solver's personal preference. Let's embark on a journey to dissect this equation and unveil its solutions.

One of the most intuitive methods for solving quadratic equations is factoring. Factoring involves breaking down the quadratic expression into a product of two linear expressions. If we can factor the quadratic expression, then we can set each factor equal to zero and solve for the variable. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. For the given equation, $3y^2 + 5y - 8 = 0$, we seek two binomials that, when multiplied, yield the original quadratic expression. This requires careful consideration of the coefficients and constants involved.

Another powerful technique is completing the square. Completing the square involves manipulating the quadratic equation to create a perfect square trinomial on one side, which can then be factored as a squared binomial. This method is particularly useful when the quadratic equation is not easily factorable. It involves a series of algebraic manipulations, including adding and subtracting constants to both sides of the equation, to transform the equation into a more manageable form. Once the perfect square trinomial is created, we can take the square root of both sides and solve for the variable.

Finally, the quadratic formula stands as a universal tool, a reliable workhorse that can solve any quadratic equation, regardless of its complexity. The quadratic formula is derived by completing the square on the general form of a quadratic equation, $ax^2 + bx + c = 0$, and provides a direct solution for the variable in terms of the coefficients a, b, and c. This formula is a powerful tool in our arsenal, guaranteeing a solution even when other methods prove challenging.

In the subsequent sections, we will delve into each of these methods, applying them to our equation $3y^2 + 5y - 8 = 0$ and comparing their approaches. By understanding the nuances of each method, we can develop a comprehensive understanding of quadratic equations and their solutions.

Method 1: Factoring the Quadratic Equation

Factoring is often the first method to consider when solving quadratic equations. It involves expressing the quadratic expression as a product of two linear factors. This method relies 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. For the equation $3y^2 + 5y - 8 = 0$, we seek to rewrite the left-hand side as a product of two binomials.

The key to factoring lies in finding two numbers that satisfy two conditions: their product must equal the product of the leading coefficient (3) and the constant term (-8), which is -24, and their sum must equal the middle coefficient (5). This can be a bit of a puzzle, requiring us to consider the factors of -24 and their possible sums. After some deliberation, we find that the numbers 8 and -3 satisfy these conditions, as 8 * -3 = -24 and 8 + (-3) = 5.

Now, we rewrite the middle term (5y) using these two numbers: $3y^2 + 8y - 3y - 8 = 0$. This step is crucial as it allows us to group the terms and factor by grouping. We group the first two terms and the last two terms: $(3y^2 + 8y) + (-3y - 8) = 0$. From the first group, we can factor out a 'y', and from the second group, we can factor out a '-1': $y(3y + 8) - 1(3y + 8) = 0$. Notice that we now have a common factor of $(3y + 8)$.

Factoring out the common factor, we get: $(3y + 8)(y - 1) = 0$. Now, we apply the zero-product property, setting each factor equal to zero: $3y + 8 = 0$ or $y - 1 = 0$. Solving these linear equations, we get y = -rac{8}{3} and $y = 1$. These are the solutions to the quadratic equation obtained by factoring.

In summary, the factoring method involves breaking down the quadratic expression into a product of two linear factors, leveraging the zero-product property to find the solutions. This method is efficient when the quadratic expression is easily factorable, offering a direct path to the roots of the equation. However, not all quadratic equations are easily factorable, and in such cases, other methods may be more suitable.

Method 2: Employing the Quadratic Formula

When factoring proves challenging or time-consuming, the quadratic formula emerges as a powerful and reliable tool. This formula provides a direct solution for any quadratic equation in the standard form $ax^2 + bx + c = 0$. The formula is given by:

y = rac{-b ext{ ± } ext{√}(b^2 - 4ac)}{2a}

where a, b, and c are the coefficients of the quadratic equation. For our equation, $3y^2 + 5y - 8 = 0$, we identify a = 3, b = 5, and c = -8. Plugging these values into the quadratic formula, we get:

y = rac{-5 ext{ ± } ext{√}(5^2 - 4 * 3 * -8)}{2 * 3}

Simplifying the expression under the square root, we have:

y = rac{-5 ext{ ± } ext{√}(25 + 96)}{6}

y = rac{-5 ext{ ± } ext{√}(121)}{6}

Since the square root of 121 is 11, we get:

y = rac{-5 ext{ ± } 11}{6}

This gives us two possible solutions:

y = rac{-5 + 11}{6} = rac{6}{6} = 1

and

y = rac{-5 - 11}{6} = rac{-16}{6} = -rac{8}{3}

Thus, the quadratic formula yields the solutions $y = 1$ and y = -rac{8}{3}, which align perfectly with the solutions obtained through factoring. The quadratic formula is a versatile method that guarantees a solution, even for equations that are difficult or impossible to factor. It is a cornerstone of quadratic equation solving, providing a reliable path to the roots regardless of the equation's complexity.

In essence, the quadratic formula stands as a testament to the power of mathematical generalization, providing a single formula that can unlock the solutions to a vast array of quadratic equations. Its application involves a straightforward substitution of coefficients, followed by arithmetic simplification, making it a readily accessible tool for solving quadratic equations.

Solutions to the Quadratic Equation

Having explored both factoring and the quadratic formula, we have arrived at the solutions to the equation $3y^2 + 5y - 8 = 0$. Both methods have yielded the same results, affirming the consistency and reliability of these techniques. The solutions are:

y = 1, -rac{8}{3}

These values of 'y' are the roots or zeros of the quadratic equation, the points where the parabola represented by the equation intersects the x-axis. The solution $y = 1$ is an integer, while y = -rac{8}{3} is an improper fraction in its simplest form. These solutions satisfy the original equation, making the expression equal to zero when substituted for 'y'.

In conclusion, we have successfully navigated the realm of quadratic equations, employing both factoring and the quadratic formula to solve the equation $3y^2 + 5y - 8 = 0$. The solutions, $y = 1$ and y = -rac{8}{3}, represent the values that satisfy the equation, showcasing the power and elegance of mathematical problem-solving. The journey through these methods has not only provided us with the solutions but also deepened our understanding of quadratic equations and their properties. The ability to solve quadratic equations is a valuable skill, applicable in various fields of mathematics and beyond.

The quadratic formula is particularly useful when the equation is not easily factorable, as it provides a direct and reliable method for finding the solutions. On the other hand, factoring can be a quicker method when the equation can be factored easily.

The solutions to the equation represent the x-intercepts of the graph of the quadratic function. These points are where the parabola crosses the x-axis, and they provide valuable information about the behavior of the function. Understanding how to solve quadratic equations is a fundamental skill in algebra and is essential for many applications in mathematics, science, and engineering.

Therefore, the solutions to the equation $3y^2 + 5y - 8 = 0$ are $y = 1$ and y = -rac{8}{3}. These solutions are expressed as an integer and an improper fraction in simplest form, respectively, fulfilling the requirements of the problem statement. The process of solving this equation has highlighted the versatility of different methods and the importance of choosing the most appropriate method for a given problem. The ability to solve quadratic equations is a cornerstone of mathematical literacy, empowering us to tackle a wide range of problems and applications.

Final Answer: $y = 1, -8/3$