Finding M Value Given Quadratic Equation Solution
In mathematics, solving quadratic equations is a fundamental skill. Quadratic equations, which take the general form ax² + bx + c = 0, often have two solutions or roots. These roots can be found using various methods, such as factoring, completing the square, or the quadratic formula. However, sometimes we are given one solution and asked to find a coefficient within the equation. This article explores how to determine the value of a coefficient, specifically 'm', in a quadratic equation when one of the solutions is provided. We will delve into the properties of quadratic equations and their roots, demonstrating a step-by-step approach to solving this type of problem. Understanding this process enhances your ability to manipulate and solve algebraic equations, which is crucial in various mathematical and real-world contexts.
To tackle the problem effectively, it's crucial to grasp the fundamentals of quadratic equations and their roots. A quadratic equation is a polynomial equation of the second degree, generally expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to solve for. The solutions to this equation are called roots or zeros, representing the values of 'x' that satisfy the equation. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of these roots is determined by the discriminant (b² - 4ac). If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root; and if it's negative, there are two complex roots.
The roots of a quadratic equation are closely related to its coefficients. According to Vieta's formulas, for a quadratic equation ax² + bx + c = 0, the sum of the roots is equal to -b/a, and the product of the roots is equal to c/a. These formulas provide a powerful tool for solving problems where relationships between the roots and coefficients are involved. In our specific scenario, where one root is given and we need to find a coefficient, Vieta's formulas, along with the concept of conjugate roots (explained below), will be instrumental in determining the value of 'm'. By understanding these fundamental concepts, we lay the groundwork for a clear and efficient solution.
When dealing with quadratic equations that have irrational or complex roots, the Conjugate Root Theorem becomes incredibly useful. This theorem states that if a polynomial equation with real coefficients has an irrational root of the form a + √b, where 'a' and 'b' are rational numbers and √b is irrational, then its conjugate a - √b is also a root of the equation. Similarly, if a complex number a + bi is a root, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1), then its complex conjugate a - bi is also a root. This theorem is crucial because it allows us to quickly identify the second root of a quadratic equation when one irrational or complex root is known, provided the coefficients of the equation are real.
In the context of our problem, we are given that 3 - √2 is a solution of the quadratic equation x² + mx + 7 = 0. Since the coefficients of this equation are real (1, 'm', and 7), and the given root is an irrational number, we can apply the Conjugate Root Theorem. This tells us that the conjugate of 3 - √2, which is 3 + √2, must also be a solution of the equation. Knowing both roots of the quadratic equation allows us to use Vieta's formulas or direct substitution to find the unknown coefficient 'm'. The Conjugate Root Theorem simplifies the process by providing us with the second root, which is essential for solving the problem efficiently.
The problem at hand presents us with a specific quadratic equation and one of its solutions. We are given the quadratic equation x² + mx + 7 = 0, where 'm' is an unknown coefficient that we need to determine. We are also told that one of the solutions, or roots, of this equation is 3 - √2. Our task is to find the value of 'm'. This type of problem often appears in algebra and requires a solid understanding of quadratic equations, their properties, and the relationships between their roots and coefficients. To solve this, we will leverage the information provided—the equation itself and the known root—along with mathematical principles such as the Conjugate Root Theorem and Vieta's formulas. By carefully applying these concepts, we can systematically deduce the value of 'm'. The problem highlights the importance of recognizing patterns and applying appropriate theorems to solve algebraic problems efficiently.
As established earlier, the Conjugate Root Theorem is instrumental in solving this problem. We know that 3 - √2 is one solution of the quadratic equation x² + mx + 7 = 0. Since the coefficients of the quadratic equation are real numbers (1, 'm', and 7), and the given root 3 - √2 is an irrational number, we can confidently apply the Conjugate Root Theorem. This theorem tells us that if 3 - √2 is a root, then its conjugate must also be a root. The conjugate of 3 - √2 is obtained by changing the sign of the irrational part, resulting in 3 + √2. Therefore, the second root of the quadratic equation is 3 + √2.
Having identified both roots of the quadratic equation, we are now in a strong position to find the value of 'm'. Knowing both roots allows us to use the relationships between the roots and coefficients of a quadratic equation, as described by Vieta's formulas. This step is crucial in moving towards the solution, as it bridges the gap between the known roots and the unknown coefficient 'm'. The application of the Conjugate Root Theorem has simplified the problem significantly by providing us with the complete set of roots, which is essential for utilizing Vieta's formulas effectively.
With both roots of the quadratic equation identified, we can now employ Vieta's formulas to find the value of 'm'. Vieta's formulas provide a direct relationship between the roots of a polynomial equation and its coefficients. For a quadratic equation in the form x² + mx + 7 = 0, Vieta's formulas state that the sum of the roots is equal to the negative of the coefficient of the 'x' term (which is '-m' in this case), and the product of the roots is equal to the constant term (which is '7' in this case).
Let the roots be denoted as r₁ and r₂. We have r₁ = 3 - √2 and r₂ = 3 + √2. According to Vieta's formulas:
- r₁ + r₂ = -m
- r₁ * r₂ = 7
We are interested in finding 'm', so we will use the first formula. Substituting the values of r₁ and r₂, we get:
(3 - √2) + (3 + √2) = -m
Simplifying the left side of the equation:
3 - √2 + 3 + √2 = 6
So, we have:
6 = -m
Multiplying both sides by -1 to solve for 'm':
m = -6
Therefore, the value of 'm' in the quadratic equation x² + mx + 7 = 0 is -6. This demonstrates how Vieta's formulas can be effectively used to find unknown coefficients in a quadratic equation when the roots are known.
To ensure the accuracy of our result, it's crucial to verify the solution. We found that m = -6. Let's substitute this value back into the original quadratic equation:
x² + mx + 7 = 0
Substituting m = -6, we get:
x² - 6x + 7 = 0
Now, we know that the roots of this equation are 3 - √2 and 3 + √2. We can verify if these roots satisfy the equation by substituting each root individually into the equation.
First, let's substitute x = 3 - √2:
(3 - √2)² - 6(3 - √2) + 7
Expanding and simplifying:
(9 - 6√2 + 2) - (18 - 6√2) + 7
11 - 6√2 - 18 + 6√2 + 7
11 - 18 + 7 - 6√2 + 6√2
0
So, x = 3 - √2 is indeed a solution.
Next, let's substitute x = 3 + √2:
(3 + √2)² - 6(3 + √2) + 7
Expanding and simplifying:
(9 + 6√2 + 2) - (18 + 6√2) + 7
11 + 6√2 - 18 - 6√2 + 7
11 - 18 + 7 + 6√2 - 6√2
0
So, x = 3 + √2 is also a solution.
Since both roots satisfy the equation x² - 6x + 7 = 0, our calculated value of m = -6 is correct. This verification step is essential to confirm the validity of the solution and ensure that no errors were made during the problem-solving process.
In this article, we successfully determined the value of 'm' in the quadratic equation x² + mx + 7 = 0, given that one of its solutions is 3 - √2. We accomplished this by leveraging the Conjugate Root Theorem to find the second root (3 + √2) and then applying Vieta's formulas to relate the roots to the coefficients of the quadratic equation. This process demonstrates a powerful approach to solving problems where one root of a quadratic equation is known, and an unknown coefficient needs to be determined.
The key takeaways from this exercise include the importance of understanding fundamental concepts such as quadratic equations, roots, and coefficients, as well as the application of specific theorems like the Conjugate Root Theorem and Vieta's formulas. The Conjugate Root Theorem allows us to quickly identify the second root when dealing with irrational or complex roots, while Vieta's formulas provide a direct link between the roots and coefficients, enabling us to solve for unknown values. Furthermore, the verification step is crucial in ensuring the accuracy of the solution.
This problem-solving approach can be applied to various similar scenarios in algebra and is a valuable skill for anyone studying mathematics. By mastering these techniques, you can confidently tackle more complex problems involving quadratic equations and their properties.
- Quadratic equation
- Conjugate Root Theorem
- Vieta's formulas
- Solving for m
- Irrational roots
- Quadratic equation solutions
- Algebra
- Mathematical problem solving
- Coefficient of quadratic equation
