Finding The Roots Of The Function Y=4x^2 + 2x - 30

Unlocking the solutions to the equation $y = 4x^2 + 2x - 30$ involves a journey into the heart of quadratic functions and their roots. In mathematical terms, finding the roots of a function means identifying the values of $x$ that make the function equal to zero. These roots, also known as zeros or x-intercepts, are the points where the graph of the function intersects the x-axis. For the given quadratic function, $y = 4x^2 + 2x - 30$, we embark on a step-by-step process to unravel these elusive values. This exploration will not only reveal the roots but also illuminate the underlying principles of quadratic equations and their solutions.

The first step in finding the roots of the function is to set $y$ equal to zero. This transforms our function into an equation: $0 = 4x^2 + 2x - 30$. This equation is the cornerstone of our quest, as its solutions directly correspond to the roots of the original function. Setting the function to zero allows us to focus solely on the values of $x$ that satisfy this condition, effectively isolating the roots from the broader context of the function's behavior. This initial step is crucial because it frames the problem in a way that allows us to apply various algebraic techniques to find the solutions.

The equation we now have, $0 = 4x^2 + 2x - 30$, is a quadratic equation in standard form, which is generally expressed as $ax^2 + bx + c = 0$. In our case, $a = 4$, $b = 2$, and $c = -30$. Recognizing the equation's structure is essential because it guides us in choosing the most appropriate method for solving it. Quadratic equations can be solved through several techniques, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and weaknesses, and the choice often depends on the specific characteristics of the equation. Factoring, for instance, is a powerful method when the quadratic expression can be easily factored into two binomials. However, when factoring is not straightforward, other methods like the quadratic formula become more suitable.

To simplify the equation and make it easier to handle, the next logical step is to factor out the greatest common factor (GCF). The GCF is the largest number that divides all the coefficients of the equation without leaving a remainder. In our equation, $4x^2 + 2x - 30$, the coefficients are 4, 2, and -30. The GCF of these numbers is 2. Factoring out 2 from the equation gives us: $0 = 2(2x^2 + x - 15)$. This step is crucial because it reduces the complexity of the quadratic expression, making subsequent steps like factoring the trinomial or applying the quadratic formula significantly easier. By removing the GCF, we essentially work with smaller numbers, which simplifies the arithmetic and reduces the chance of errors. This simplification is a common strategy in algebra, allowing us to break down complex problems into more manageable parts.

After factoring out the GCF, we are left with the simplified equation $0 = 2(2x^2 + x - 15)$. Our focus now shifts to factoring the trinomial $2x^2 + x - 15$. Factoring a trinomial involves expressing it as the product of two binomials. This process requires careful consideration of the coefficients and constants involved. In this case, we are looking for two binomials of the form $(Ax + B)(Cx + D)$ such that when multiplied, they yield $2x^2 + x - 15$. The key is to find the correct combination of factors that satisfy the conditions imposed by the coefficients of the trinomial. There are several strategies for factoring trinomials, including trial and error, the AC method, and grouping. The choice of method often depends on personal preference and the specific characteristics of the trinomial. The goal is to systematically explore different combinations of factors until the correct pair is found.

The trinomial $2x^2 + x - 15$ can be factored into $(2x - 5)(x + 3)$. This factorization is a critical step in solving the quadratic equation because it transforms the equation from a sum of terms into a product of factors. The factored equation is: $0 = 2(2x - 5)(x + 3)$. This form is particularly useful because it directly leads to the roots of the equation through the zero-product property. The zero-product property states that if the product of several factors is zero, then at least one of the factors must be zero. This property is a cornerstone of solving equations by factoring, as it allows us to break down a complex equation into simpler ones.

Therefore, the equation becomes $0 = 2(2x - 5)(x + 3)$.

After factoring the quadratic expression, the final step in finding the roots is to apply the zero-product property. This property, a fundamental principle in algebra, states that if the product of several factors is equal to zero, then at least one of the factors must be equal to zero. In our case, the factored equation is $0 = 2(2x - 5)(x + 3)$. This equation represents the product of three factors: 2, $(2x - 5)$, and $(x + 3)$. According to the zero-product property, for this product to be zero, at least one of these factors must be zero. Since 2 is a constant and cannot be zero, we focus on the other two factors.

To find the roots, we set each of the variable factors equal to zero and solve for $x$. This gives us two separate equations: $2x - 5 = 0$ and $x + 3 = 0$. Each of these equations is a simple linear equation that can be easily solved by isolating $x$. Solving these equations will give us the values of $x$ that make the original quadratic function equal to zero, which are precisely the roots we are seeking. This process demonstrates the power of factoring in simplifying complex equations and making their solutions readily accessible. The zero-product property acts as a bridge, connecting the factored form of the equation to its solutions.

Let's solve the first equation, $2x - 5 = 0$. To isolate $x$, we first add 5 to both sides of the equation, which gives us $2x = 5$. Then, we divide both sides by 2, resulting in x = rac{5}{2}. This value, rac{5}{2}, is one of the roots of the quadratic function. It represents a point where the graph of the function intersects the x-axis. In the context of the original problem, this means that when $x$ is equal to rac{5}{2}, the value of the function $y = 4x^2 + 2x - 30$ is zero. This root is a crucial piece of information about the function's behavior and its relationship to the x-axis.

Now, let's solve the second equation, $x + 3 = 0$. To isolate $x$, we simply subtract 3 from both sides of the equation, which gives us $x = -3$. This value, -3, is the second root of the quadratic function. Like the first root, it represents a point where the graph of the function intersects the x-axis. When $x$ is equal to -3, the value of the function $y = 4x^2 + 2x - 30$ is zero. This second root completes our solution set, providing a comprehensive understanding of the function's zeros.

Therefore, the roots of the function $y = 4x^2 + 2x - 30$ are x = rac{5}{2} and $x = -3$. These roots are the solutions to the equation $0 = 4x^2 + 2x - 30$, and they represent the x-intercepts of the graph of the function. Understanding how to find roots is a fundamental skill in algebra and calculus, with applications in various fields, including physics, engineering, and economics. The process we have followed, from setting the function to zero to factoring and applying the zero-product property, is a classic example of how algebraic techniques can be used to solve real-world problems.

In summary, finding the roots of a quadratic function involves a systematic approach that begins with setting the function equal to zero. Factoring out the greatest common factor simplifies the equation, making it easier to factor the trinomial. The zero-product property then allows us to find the roots by setting each factor equal to zero and solving for $x$. These roots are the values of $x$ that make the function equal to zero, and they represent the points where the graph of the function intersects the x-axis. The roots of the function $y = 4x^2 + 2x - 30$ are rac{5}{2} and $-3$, providing a complete solution to the problem.