Finding The Second Solution Of A Quadratic Equation Given One Solution

In mathematics, solving equations is a fundamental skill. Quadratic equations, in particular, appear in various contexts, from physics to engineering. 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 ≠ 0. These equations can have two, one, or no real solutions, often called roots or zeros. This article will explore how to find the second solution of a quadratic equation when one solution is already known. Let's delve into the methods and techniques to solve such problems effectively.

Before diving into the problem at hand, let's recap the basics of quadratic equations. A quadratic equation can be solved using various methods, including factoring, completing the square, and the quadratic formula. Each method has its strengths and is suitable for different types of equations.

• Factoring: This method involves breaking down the quadratic expression into two linear factors. For example, x² - 5x + 6 = 0 can be factored into (x - 2)(x - 3) = 0, giving solutions x = 2 and x = 3. Factoring is efficient when the roots are rational numbers.
• Completing the Square: This method transforms the quadratic equation into a perfect square trinomial. It's especially useful for deriving the quadratic formula and solving equations that don't factor easily. The process involves adding and subtracting a term to make the expression a perfect square.
• Quadratic Formula: This formula provides a direct way to find the solutions of any quadratic equation. Given ax² + bx + c = 0, the solutions are given by x = (-b ± √(b² - 4ac)) / (2a). The quadratic formula is a powerful tool, especially when the equation is not easily factorable.

The discriminant, b² - 4ac, determines the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If it is negative, there are no real roots, but there are two complex roots.

Sonya has encountered a problem where she knows one solution of a quadratic equation. The equation is given in the form (6x + 4)² = □, and one solution is x = -4/3. Our task is to find the other solution. This problem involves understanding the structure of quadratic equations and using algebraic manipulation to arrive at the solution. Knowing one solution provides a significant advantage, as we can use this information to deduce the other solution more efficiently.

To find the other solution, we first need to rewrite the equation in a standard quadratic form. The given equation is (6x + 4)² = □. Let's denote the unknown value on the right-hand side as k. So, the equation becomes (6x + 4)² = k. We know that x = -4/3 is a solution. Plugging this value into the equation, we get:

(6(-4/3) + 4)² = k (-8 + 4)² = k (-4)² = k 16 = k

So, the equation is (6x + 4)² = 16. Now, we need to solve this equation for x. We can start by taking the square root of both sides:

√(6x + 4)² = ±√16 6x + 4 = ±4

This gives us two separate equations:

1. 6x + 4 = 4
2. 6x + 4 = -4

Solving the first equation:

6x + 4 = 4 6x = 0 x = 0

Solving the second equation:

6x + 4 = -4 6x = -8 x = -8/6 x = -4/3

We already knew that x = -4/3 is one solution. Therefore, the other solution is x = 0.

Another way to approach this problem is by expanding the original equation and rearranging it into the standard quadratic form. The equation is (6x + 4)² = 16. Expanding the left side, we get:

(6x + 4)(6x + 4) = 16 36x² + 48x + 16 = 16

Subtracting 16 from both sides, we have:

36x² + 48x = 0

We can factor out a common factor of 12x:

12x(3x + 4) = 0

This gives us two possible solutions:

1. 12x = 0 => x = 0
2. 3x + 4 = 0 => 3x = -4 => x = -4/3

Again, we find that the solutions are x = 0 and x = -4/3. This method confirms our previous result and provides an alternative way to solve the equation.

Understanding different methods for solving quadratic equations is crucial. Each method—factoring, completing the square, and the quadratic formula—has its advantages and can be more efficient depending on the specific equation. In this case, recognizing the structure of the equation (6x + 4)² = 16 allowed us to take a shortcut by taking the square root directly. If we had not recognized this, we could still solve it by expanding and factoring, but it would have involved more steps.

Furthermore, knowing that one solution is x = -4/3 helped us verify our results and ensure that our calculations were correct. This highlights the importance of checking solutions, especially in more complex problems.

In conclusion, we found the other solution of the equation (6x + 4)² = 16, given that one solution is x = -4/3. By setting up the equation, substituting one known solution, and solving for the other, we determined that the other solution is x = 0. We also verified this result by expanding the equation and factoring it. This exercise demonstrates the importance of understanding quadratic equations and the various methods for solving them. Whether through direct algebraic manipulation or by recognizing specific patterns, having a solid foundation in these concepts is essential for tackling more advanced mathematical problems.

This problem also underscores the significance of approaching mathematical problems with flexibility and choosing the most efficient method for the given situation. By combining different techniques and verifying our solutions, we can confidently solve quadratic equations and other mathematical challenges.

Mastering quadratic equations is a cornerstone of algebra and is crucial for success in higher mathematics. The ability to solve these equations using different methods—factoring, completing the square, the quadratic formula, and algebraic manipulation—is an invaluable skill. By understanding the underlying principles and practicing regularly, students can develop the confidence to tackle even the most challenging problems. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing critical thinking skills. Keep practicing, and you'll continue to improve your mathematical abilities.