In the realm of mathematics, quadratic equations hold a significant position. These equations, characterized by the presence of a squared variable, frequently arise in various scientific and engineering disciplines. In this comprehensive guide, we will embark on a journey to solve the quadratic equation 3y² + 7y = 6. We will dissect the equation, transform it into its standard form, and then employ the powerful quadratic formula to arrive at the solutions. Along the way, we will delve into the intricacies of quadratic equations, ensuring a thorough understanding of the underlying concepts.
Transforming the Equation into Standard Form
Our initial task is to transform the given equation, 3y² + 7y = 6, into its standard form. The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' represent constant coefficients, and 'x' is the variable. To achieve this transformation, we need to manipulate the equation algebraically.
Let's begin by subtracting 6 from both sides of the equation:
3y² + 7y - 6 = 6 - 6
This simplifies to:
3y² + 7y - 6 = 0
Now, we have successfully transformed the equation into the standard form ax² + bx + c = 0, where:
- a = 3
- b = 7
- c = -6
This standard form is crucial because it allows us to directly apply the quadratic formula, a cornerstone in solving quadratic equations.
Unveiling the Quadratic Formula: A Key to Solutions
The quadratic formula is a mathematical expression that provides the solutions to any quadratic equation expressed in standard form. This formula is derived from the method of completing the square and is a fundamental tool in algebra. The quadratic formula is expressed as follows:
y = (-b ± √(b² - 4ac)) / (2a)
Where:
- y represents the solutions (also known as roots) of the quadratic equation.
- a, b, and c are the coefficients of the quadratic equation in standard form (ax² + bx + c = 0).
The ± symbol indicates that there are potentially two solutions: one obtained by adding the square root term and the other by subtracting it.
Applying the Quadratic Formula to Our Equation
Now that we have the quadratic formula at our disposal, let's apply it to our equation, 3y² + 7y - 6 = 0. We have already identified the coefficients as:
- a = 3
- b = 7
- c = -6
Substituting these values into the quadratic formula, we get:
y = (-7 ± √(7² - 4 * 3 * -6)) / (2 * 3)
Let's simplify this expression step-by-step:
y = (-7 ± √(49 + 72)) / 6
y = (-7 ± √121) / 6
y = (-7 ± 11) / 6
Now, we have two possible solutions:
- y = (-7 + 11) / 6 = 4 / 6 = 2 / 3
- y = (-7 - 11) / 6 = -18 / 6 = -3
Therefore, the solutions to the quadratic equation 3y² + 7y = 6 are y = 2/3 and y = -3.
Verifying the Solutions: Ensuring Accuracy
To ensure the accuracy of our solutions, we can substitute each value of 'y' back into the original equation, 3y² + 7y = 6, and verify if the equation holds true.
Let's start with y = 2/3:
3(2/3)² + 7(2/3) = 6
3(4/9) + 14/3 = 6
4/3 + 14/3 = 6
18/3 = 6
6 = 6
The equation holds true for y = 2/3, confirming it as a valid solution.
Now, let's verify y = -3:
3(-3)² + 7(-3) = 6
3(9) - 21 = 6
27 - 21 = 6
6 = 6
The equation also holds true for y = -3, further validating our solutions.
Exploring the Discriminant: Unveiling the Nature of Solutions
In the realm of quadratic equations, the discriminant plays a pivotal role in determining the nature of the solutions. The discriminant is the expression under the square root in the quadratic formula, represented as b² - 4ac. The value of the discriminant provides valuable insights into the number and type of solutions a quadratic equation possesses.
There are three possible scenarios:
- Discriminant > 0: The quadratic equation has two distinct real solutions. This signifies that the parabola represented by the quadratic equation intersects the x-axis at two distinct points.
- Discriminant = 0: The quadratic equation has exactly one real solution (a repeated root). In this case, the parabola touches the x-axis at exactly one point.
- Discriminant < 0: The quadratic equation has no real solutions. The solutions are complex numbers, indicating that the parabola does not intersect the x-axis.
For our equation, 3y² + 7y - 6 = 0, the discriminant is:
b² - 4ac = 7² - 4 * 3 * -6 = 49 + 72 = 121
Since the discriminant is greater than 0, we can confirm that our equation has two distinct real solutions, which we have already found to be y = 2/3 and y = -3.
Conclusion: Mastering the Art of Solving Quadratic Equations
In this comprehensive guide, we have embarked on a journey to solve the quadratic equation 3y² + 7y = 6. We successfully transformed the equation into its standard form, applied the quadratic formula, and arrived at the solutions y = 2/3 and y = -3. We further validated these solutions by substituting them back into the original equation. Additionally, we delved into the concept of the discriminant, gaining insights into the nature of solutions based on its value.
Mastering the art of solving quadratic equations is a fundamental skill in mathematics. The quadratic formula serves as a powerful tool, enabling us to tackle a wide range of problems in various disciplines. By understanding the underlying concepts and practicing diligently, you can confidently navigate the world of quadratic equations and their applications.
Key Takeaways:
- Quadratic equations are equations of the form ax² + bx + c = 0.
- The quadratic formula is used to find the solutions of quadratic equations.
- The discriminant (b² - 4ac) determines the nature of the solutions.
- If the discriminant is positive, there are two distinct real solutions.
- If the discriminant is zero, there is one real solution (a repeated root).
- If the discriminant is negative, there are no real solutions (complex solutions).
Further Exploration:
- Explore different methods for solving quadratic equations, such as factoring and completing the square.
- Investigate the applications of quadratic equations in real-world scenarios, such as projectile motion and optimization problems.
- Delve into the concept of complex numbers and their role in solving quadratic equations with negative discriminants.
By continuing your exploration of quadratic equations, you will solidify your understanding of this fundamental mathematical concept and enhance your problem-solving skills.
