Solving Y² = 3y + 10 By Factoring A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of solving the quadratic equation $y^2 = 3y + 10$ by factoring. Factoring is a fundamental technique in algebra that allows us to break down complex expressions into simpler ones, making it easier to find the solutions (also known as roots or zeros) of the equation. This method is particularly effective for quadratic equations that can be factored neatly. Understanding how to solve quadratic equations by factoring is crucial for various mathematical and real-world applications. This article will provide a step-by-step approach, ensuring clarity and ease of understanding for anyone looking to master this essential skill. We will not only solve the given equation but also discuss the underlying principles and concepts, enhancing your overall understanding of quadratic equations and factoring techniques. So, let's embark on this mathematical journey and unlock the secrets of factoring!

Understanding Quadratic Equations

Before we dive into solving the equation $y^2 = 3y + 10$, it's essential to understand what quadratic equations are and their standard form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The coefficient 'a' cannot be zero; otherwise, the equation would be linear, not quadratic. Understanding this basic form is the cornerstone for solving these equations.

In our specific equation, $y^2 = 3y + 10$, we can see that it involves the square of the variable 'y', which confirms it as a quadratic equation. To effectively solve it by factoring, our first step will be to rearrange the equation into the standard form. This involves moving all terms to one side of the equation, leaving zero on the other side. This rearrangement allows us to clearly identify the coefficients and constant term, which are crucial for the factoring process. By recognizing the standard form, we can apply various techniques, including factoring, to find the values of 'y' that satisfy the equation. This foundational understanding of quadratic equations sets the stage for a successful problem-solving approach.

Step 1: Rearrange the Equation

The first crucial step in solving the equation $y^2 = 3y + 10$ by factoring is to rearrange it into the standard quadratic form, which is $ay^2 + by + c = 0$. This form allows us to clearly identify the coefficients and the constant term, making the factoring process more straightforward. To rearrange our equation, we need to move all terms to one side, leaving zero on the other side. This is achieved by subtracting $3y$ and $10$ from both sides of the equation. Performing this operation ensures that the equation remains balanced and maintains its original solutions.

Starting with $y^2 = 3y + 10$, we subtract $3y$ from both sides to get $y^2 - 3y = 10$. Next, we subtract $10$ from both sides to obtain the equation in standard form: $y^2 - 3y - 10 = 0$. Now, we can clearly see that the equation is in the form $ay^2 + by + c = 0$, where $a = 1$, $b = -3$, and $c = -10$. This rearranged form is essential for the subsequent steps in the factoring process. By having the equation in this standard format, we can easily apply factoring techniques to find the solutions for 'y'. This step is the foundation upon which the rest of the solution is built, and mastering it is key to solving quadratic equations.

Step 2: Factor the Quadratic Expression

With the equation now in standard form, $y^2 - 3y - 10 = 0$, the next step is to factor the quadratic expression. Factoring involves breaking down the quadratic expression into two binomials, which, when multiplied together, give us the original expression. This process relies on finding two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. In our case, we need to find two numbers that multiply to the constant term (-10) and add up to the coefficient of the linear term (-3). This is a critical step, and understanding how to identify these numbers is essential for successful factoring.

To find these numbers, we can consider the factors of -10. These include pairs such as (1, -10), (-1, 10), (2, -5), and (-2, 5). We then check which pair adds up to -3. Looking at these pairs, we find that the numbers 2 and -5 satisfy both conditions: $2 * -5 = -10$ and $2 + (-5) = -3$. Once we've identified these numbers, we can rewrite the quadratic expression as a product of two binomials. The binomials will have the form $(y + p)(y + q)$, where 'p' and 'q' are the numbers we found. In our case, this translates to $(y + 2)(y - 5)$. Thus, the factored form of the quadratic expression is $(y + 2)(y - 5) = 0$. This factored form is crucial because it allows us to use the zero-product property to find the solutions for 'y'.

Step 3: Apply the Zero-Product Property

Now that we have factored the quadratic equation into the form $(y + 2)(y - 5) = 0$, the next crucial step is to apply the zero-product property. This fundamental property states that if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if $A * B = 0$, then either $A = 0$ or $B = 0$ (or both). This property is the key to finding the solutions of our quadratic equation once it is in factored form. By applying the zero-product property, we can transform the single equation into two simpler equations, each of which can be easily solved for 'y'.

In our case, the factors are $(y + 2)$ and $(y - 5)$. Applying the zero-product property means we set each factor equal to zero, resulting in two separate equations: $y + 2 = 0$ and $y - 5 = 0$. These are linear equations, which are much easier to solve than the original quadratic equation. This step is a direct application of a core algebraic principle and is essential for transitioning from the factored form to the individual solutions. By understanding and correctly applying the zero-product property, we can effectively break down the problem into manageable parts and find the values of 'y' that satisfy the original equation.

Step 4: Solve for y

After applying the zero-product property, we now have two simple linear equations: $y + 2 = 0$ and $y - 5 = 0$. The next step is to solve each of these equations for 'y'. Solving for a variable involves isolating it on one side of the equation. This is achieved by performing the same operation on both sides of the equation to maintain the balance. These operations are based on fundamental algebraic principles and are crucial for finding the values of 'y' that make each equation true.

For the first equation, $y + 2 = 0$, we can isolate 'y' by subtracting 2 from both sides. This gives us $y + 2 - 2 = 0 - 2$, which simplifies to $y = -2$. This means that when 'y' is -2, the equation $y + 2 = 0$ is satisfied. Similarly, for the second equation, $y - 5 = 0$, we can isolate 'y' by adding 5 to both sides. This gives us $y - 5 + 5 = 0 + 5$, which simplifies to $y = 5$. This means that when 'y' is 5, the equation $y - 5 = 0$ is also satisfied. Therefore, we have found two values for 'y' that make the original factored equation true. These values are the solutions to the quadratic equation. This step is the culmination of the factoring process, where we finally determine the numerical values of the variable that satisfy the original equation.

Solution

Having solved the two linear equations derived from the zero-product property, we have found the solutions for the quadratic equation $y^2 = 3y + 10$. The solutions are the values of 'y' that make the equation true. From the equation $y + 2 = 0$, we found that $y = -2$, and from the equation $y - 5 = 0$, we found that $y = 5$. These are the two values that, when substituted back into the original equation, will satisfy the equality. Therefore, the solutions to the equation $y^2 = 3y + 10$ are $y = -2$ and $y = 5$.

To present the solutions in the requested format, we separate them with a comma. Thus, the final answer is "-2,5". This concise representation clearly communicates the two distinct values of 'y' that satisfy the given quadratic equation. It is essential to present the solutions clearly and accurately to ensure the problem is fully resolved. This final step is the culmination of the entire process, from rearranging the equation to factoring and applying the zero-product property. The result showcases the power of factoring as a method for solving quadratic equations and highlights the importance of each step in the process.

Conclusion

In conclusion, we have successfully solved the quadratic equation $y^2 = 3y + 10$ by factoring. This process involved several key steps, starting with rearranging the equation into standard form ($y^2 - 3y - 10 = 0$), then factoring the quadratic expression into two binomials $((y + 2)(y - 5) = 0$), applying the zero-product property to set each factor equal to zero, and finally solving the resulting linear equations to find the solutions ($y = -2$ and $y = 5$). This step-by-step approach demonstrates the effectiveness of factoring as a method for solving quadratic equations, particularly those that can be factored neatly.

Throughout this guide, we have emphasized the importance of understanding the underlying principles and concepts, such as the standard form of a quadratic equation, the factoring process, and the zero-product property. Mastering these concepts not only allows us to solve specific equations but also provides a solid foundation for tackling more complex algebraic problems. Factoring is a fundamental skill in algebra, and its applications extend to various areas of mathematics and real-world problem-solving. By understanding and practicing this technique, individuals can enhance their mathematical abilities and develop a deeper appreciation for the elegance and power of algebra. The solutions, $y = -2$ and $y = 5$, represent the points where the quadratic function intersects the x-axis, further illustrating the connection between algebraic solutions and graphical representations.