Solving 3x² + 5x - 2 = 0 A Step-by-Step Guide
In this comprehensive guide, we will meticulously walk through the process of solving the quadratic equation 3x² + 5x - 2 = 0. Quadratic equations, characterized by their highest power of 2, frequently appear in various fields of mathematics, physics, and engineering. Mastering the techniques to solve them is crucial for anyone seeking a deeper understanding of these disciplines. We will explore multiple methods, including factoring, the quadratic formula, and completing the square, to provide a thorough understanding of how to find the solutions (also known as roots) of this equation. This detailed explanation will not only provide the answer but also equip you with the knowledge to tackle similar quadratic equations confidently.
Understanding Quadratic Equations
Before diving into the specific solution, let's solidify our understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation, the values of x that satisfy the equation, are called roots or zeros. These roots represent the points where the parabola defined by the equation intersects the x-axis. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots is determined by the discriminant, which is the expression b² - 4ac. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; and if it is negative, there are two complex roots. Understanding these fundamental concepts lays the groundwork for effectively solving quadratic equations.
Method 1: Factoring the Quadratic Equation
One of the most efficient methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two binomials. This method is particularly useful when the quadratic equation has integer roots and the coefficients are relatively small. The goal is to rewrite the equation 3x² + 5x - 2 = 0 in the form (px + q)(rx + s) = 0, where p, q, r, and s are constants. To factor the equation 3x² + 5x - 2 = 0, we need to find two numbers that multiply to (3)(-2) = -6 and add up to 5. These numbers are 6 and -1. Now, we rewrite the middle term using these numbers: 3x² + 6x - x - 2 = 0. Next, we factor by grouping: 3x(x + 2) - 1(x + 2) = 0. Notice that (x + 2) is a common factor. Factoring it out, we get (3x - 1)(x + 2) = 0. Setting each factor equal to zero, we have 3x - 1 = 0 or x + 2 = 0. Solving these linear equations gives us the solutions x = 1/3 and x = -2. Therefore, the roots of the quadratic equation 3x² + 5x - 2 = 0 are 1/3 and -2. Factoring, when applicable, provides a straightforward approach to solving quadratic equations.
Method 2: Applying the Quadratic Formula
When factoring proves challenging or impossible, the quadratic formula provides a universally applicable method for solving quadratic equations. The quadratic formula is derived from the process of completing the square and can be used to find the roots of any quadratic equation in the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In our equation, 3x² + 5x - 2 = 0, we identify the coefficients as a = 3, b = 5, and c = -2. Substituting these values into the quadratic formula, we get:
x = (-5 ± √(5² - 4(3)(-2))) / (2(3))
Simplifying the expression under the square root:
x = (-5 ± √(25 + 24)) / 6
x = (-5 ± √49) / 6
Since √49 = 7, we have:
x = (-5 ± 7) / 6
This gives us two possible solutions:
x₁ = (-5 + 7) / 6 = 2 / 6 = 1/3
x₂ = (-5 - 7) / 6 = -12 / 6 = -2
Thus, the solutions to the quadratic equation 3x² + 5x - 2 = 0, using the quadratic formula, are x = 1/3 and x = -2. The quadratic formula ensures we can solve any quadratic equation, regardless of the complexity of its coefficients or the nature of its roots.
Method 3: Completing the Square
Completing the square is another powerful technique for solving quadratic equations. This method involves manipulating the equation into a form where one side is a perfect square trinomial. While it may seem more complex than factoring or the quadratic formula at first, it provides valuable insight into the structure of quadratic equations and is particularly useful in certain situations, such as deriving the quadratic formula itself. To solve 3x² + 5x - 2 = 0 by completing the square, we first divide the entire equation by the leading coefficient, 3, to make the coefficient of x² equal to 1:
x² + (5/3)x - (2/3) = 0
Next, we move the constant term to the right side of the equation:
x² + (5/3)x = 2/3
Now, we complete the square on the left side. To do this, we take half of the coefficient of the x term, square it, and add it to both sides of the equation. Half of (5/3) is (5/6), and squaring it gives us (25/36). So, we add (25/36) to both sides:
x² + (5/3)x + (25/36) = 2/3 + 25/36
The left side is now a perfect square trinomial, which can be factored as:
(x + 5/6)² = 2/3 + 25/36
To simplify the right side, we find a common denominator:
(x + 5/6)² = 24/36 + 25/36 = 49/36
Now, we take the square root of both sides:
x + 5/6 = ±√(49/36) = ±7/6
This gives us two equations:
x + 5/6 = 7/6 and x + 5/6 = -7/6
Solving for x in each case:
x = 7/6 - 5/6 = 2/6 = 1/3
x = -7/6 - 5/6 = -12/6 = -2
Thus, the solutions to the quadratic equation 3x² + 5x - 2 = 0, obtained by completing the square, are x = 1/3 and x = -2. Completing the square provides a deeper understanding of the structure of quadratic equations and confirms the solutions we found using factoring and the quadratic formula.
Verification of Solutions
To ensure the accuracy of our solutions, it is crucial to verify them by substituting them back into the original equation. This step confirms that the values we obtained indeed satisfy the equation and are the correct roots. Let's verify the solutions x = 1/3 and x = -2 in the equation 3x² + 5x - 2 = 0.
For x = 1/3:
3(1/3)² + 5(1/3) - 2 = 3(1/9) + 5/3 - 2 = 1/3 + 5/3 - 2 = 6/3 - 2 = 2 - 2 = 0
The equation holds true for x = 1/3.
For x = -2:
3(-2)² + 5(-2) - 2 = 3(4) - 10 - 2 = 12 - 10 - 2 = 2 - 2 = 0
The equation also holds true for x = -2.
Since both values satisfy the original equation, we can confidently conclude that the solutions to 3x² + 5x - 2 = 0 are x = 1/3 and x = -2. Verification is a critical step in problem-solving, particularly in mathematics, to ensure the accuracy and validity of the results.
Conclusion
In this comprehensive guide, we explored various methods for solving the quadratic equation 3x² + 5x - 2 = 0. We successfully employed factoring, the quadratic formula, and completing the square to arrive at the solutions x = 1/3 and x = -2. Each method provides a unique approach to tackling quadratic equations, and understanding these techniques empowers you to solve a wide range of problems. Factoring offers an efficient solution when applicable, the quadratic formula provides a universal method for any quadratic equation, and completing the square offers a deeper understanding of the equation's structure. Furthermore, we emphasized the importance of verifying solutions to ensure their accuracy. By mastering these methods and practicing regularly, you can confidently solve quadratic equations and apply this knowledge to various mathematical and real-world problems. Quadratic equations are a fundamental concept in mathematics, and proficiency in solving them is a valuable skill in numerous disciplines.
