Solving For Coefficients In Quadratic Equations When M+n=mn=3

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In the fascinating realm of mathematics, quadratic equations hold a special place. These equations, characterized by their highest power of 2, often present intriguing puzzles that require a blend of algebraic manipulation and insightful reasoning to solve. In this article, we embark on a journey to unravel one such puzzle, where we are tasked with determining the values of 'a' and 'c' in a quadratic equation, given specific information about its zeroes.

The Quadratic Equation and Its Zeroes

Before we delve into the heart of the problem, let's take a moment to refresh our understanding of quadratic equations and their zeroes. A quadratic equation is generally represented in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The zeroes of a quadratic equation are the values of 'x' that make the equation true, or in other words, the values of 'x' that make the expression ax² + bx + c equal to zero. These zeroes are also known as the roots of the equation.

A fundamental concept in the world of quadratic equations is the relationship between the zeroes and the coefficients of the equation. For a quadratic equation ax² + bx + c = 0, if 'm' and 'n' are the zeroes, then the following relationships hold true:

  • Sum of zeroes (m + n) = -b/a
  • Product of zeroes (m n) = c/a

These relationships provide us with a powerful tool to connect the zeroes of a quadratic equation with its coefficients, enabling us to solve a variety of problems.

The Problem at Hand

Now, let's turn our attention to the specific problem we aim to solve. We are given the quadratic equation ax² – 12x + c = 0, and we are told that 'm' and 'n' are its zeroes. Furthermore, we are provided with the crucial information that m + n = m n = 3. Our mission is to determine the values of 'a' and 'c' that satisfy these conditions.

Cracking the Code: Utilizing the Relationships

To solve this problem, we will leverage the relationships between the zeroes and the coefficients of a quadratic equation that we discussed earlier. We know that:

  • m + n = -(-12)/a = 12/a
  • m n = c/a

We are also given that m + n = 3 and m n = 3. Now, we can substitute these values into the equations above:

  • 3 = 12/a
  • 3 = c/a

From the first equation, we can easily solve for 'a':

  • a = 12/3 = 4

Now that we have the value of 'a', we can substitute it into the second equation to find 'c':

  • 3 = c/4
  • c = 3 * 4 = 12

Therefore, the values of 'a' and 'c' that satisfy the given conditions are a = 4 and c = 12.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify our findings. Let's substitute the values of 'a' and 'c' back into the original quadratic equation:

  • 4x² – 12x + 12 = 0

Now, let's find the zeroes of this equation. We can divide the entire equation by 4 to simplify it:

  • x² – 3x + 3 = 0

To find the zeroes, we can use the quadratic formula:

  • x = (-b ± √(b² – 4ac)) / 2a

In this case, a = 1, b = -3, and c = 3. Substituting these values into the quadratic formula, we get:

  • x = (3 ± √((-3)² – 4 * 1 * 3)) / 2 * 1
  • x = (3 ± √(-3)) / 2
  • x = (3 ± i√3) / 2

The zeroes of the equation are complex numbers: (3 + i√3) / 2 and (3 - i√3) / 2. Let's denote these zeroes as 'm' and 'n', respectively.

Now, let's check if m + n = 3 and m n = 3:

  • m + n = (3 + i√3) / 2 + (3 - i√3) / 2 = 3
  • m n = ((3 + i√3) / 2) * ((3 - i√3) / 2) = (9 + 3) / 4 = 3

As we can see, the conditions m + n = 3 and m n = 3 are indeed satisfied. This confirms that our solution, a = 4 and c = 12, is correct.

A Deeper Dive: Exploring the Significance

This problem serves as a great example of how the relationships between the zeroes and coefficients of a quadratic equation can be used to solve problems. By understanding these relationships, we can unravel the mysteries hidden within quadratic equations and gain a deeper appreciation for the beauty and power of algebra.

Moreover, this problem highlights the importance of verifying our solutions. By substituting the values we obtained back into the original equation and checking if the given conditions are met, we can ensure the accuracy of our work and avoid potential errors.

In conclusion, we have successfully determined the values of 'a' and 'c' in the quadratic equation ax² – 12x + c = 0, given that 'm' and 'n' are its zeroes and m + n = m n = 3. The values we found are a = 4 and c = 12. This problem underscores the significance of the relationships between the zeroes and coefficients of a quadratic equation and the importance of verifying our solutions.

Expanding the Horizon: Further Exploration

If you found this problem intriguing, there's a whole world of quadratic equations waiting to be explored. You can delve deeper into topics such as:

  • The discriminant: The discriminant (b² – 4ac) of a quadratic equation provides valuable information about the nature of its roots (real, complex, distinct, or repeated).
  • Completing the square: This technique can be used to rewrite a quadratic equation in a form that makes it easier to solve.
  • Applications of quadratic equations: Quadratic equations have numerous applications in various fields, such as physics, engineering, and economics.

By venturing into these areas, you can further enhance your understanding of quadratic equations and their significance in the world of mathematics and beyond.

Quadratic equations, a cornerstone of algebra, often present intriguing challenges. These equations, characterized by the highest power of 2, demand a blend of algebraic dexterity and insightful reasoning to solve. In this article, we'll dissect a specific problem where we aim to pinpoint the values of 'a' and 'c' within a quadratic equation, armed with specific details about its zeroes.

Quadratic Equations and Zeroes: A Primer

Before tackling the core of the problem, let's quickly recap quadratic equations and their zeroes. A quadratic equation typically takes the form ax² + bx + c = 0, where 'a', 'b', and 'c' represent constants, and 'x' is the variable. The zeroes of a quadratic equation are the 'x' values that make the equation true – essentially, the 'x' values that make ax² + bx + c equal zero. These zeroes are also referred to as the roots of the equation.

A key principle in quadratic equations is the link between the zeroes and the equation's coefficients. For a quadratic equation ax² + bx + c = 0, where 'm' and 'n' are the zeroes, the following relationships are fundamental:

  • Zero Sum (m + n) = -b/a
  • Zero Product (m n) = c/a

These relationships offer a robust tool for connecting the zeroes of a quadratic equation with its coefficients, enabling us to tackle various problems effectively.

The Challenge: Solving for 'a' and 'c'

Let's focus on the specific problem we're addressing. We're given the quadratic equation ax² – 12x + c = 0. We know that 'm' and 'n' are its zeroes, and we have the crucial information that m + n = m n = 3. Our objective is to determine the 'a' and 'c' values that satisfy these conditions.

Unlocking the Solution: Applying the Relationships

To crack this problem, we'll harness the relationships between a quadratic equation's zeroes and coefficients, as discussed earlier. We know:

  • m + n = -(-12)/a = 12/a
  • m n = c/a

We're also given that m + n = 3 and m n = 3. Substituting these values into the equations yields:

  • 3 = 12/a
  • 3 = c/a

From the first equation, solving for 'a' is straightforward:

  • a = 12/3 = 4

Now, with 'a' in hand, we can plug it into the second equation to find 'c':

  • 3 = c/4
  • c = 3 * 4 = 12

Therefore, the values of 'a' and 'c' that meet the given conditions are a = 4 and c = 12.

Solution Verification

To ensure our solution's accuracy, let's verify our findings. We'll substitute the 'a' and 'c' values back into the original quadratic equation:

  • 4x² – 12x + 12 = 0

Now, let's determine the zeroes of this equation. Dividing the equation by 4 simplifies it:

  • x² – 3x + 3 = 0

Using the quadratic formula to find the zeroes:

  • x = (-b ± √(b² – 4ac)) / 2a

Here, a = 1, b = -3, and c = 3. Plugging these values into the quadratic formula:

  • x = (3 ± √((-3)² – 4 * 1 * 3)) / 2 * 1
  • x = (3 ± √(-3)) / 2
  • x = (3 ± i√3) / 2

The equation's zeroes are complex numbers: (3 + i√3) / 2 and (3 - i√3) / 2. Let's label these zeroes as 'm' and 'n', respectively.

Checking if m + n = 3 and m n = 3:

  • m + n = (3 + i√3) / 2 + (3 - i√3) / 2 = 3
  • m n = ((3 + i√3) / 2) * ((3 - i√3) / 2) = (9 + 3) / 4 = 3

As demonstrated, the conditions m + n = 3 and m n = 3 are indeed satisfied, confirming that our solution a = 4 and c = 12 is correct.

In Conclusion

In summary, we've successfully found the values of 'a' and 'c' in the quadratic equation ax² – 12x + c = 0, given that 'm' and 'n' are its zeroes and m + n = m n = 3. The solution is a = 4 and c = 12. This exercise highlights the importance of the relationship between zeroes and coefficients in a quadratic equation, as well as the significance of verifying solutions.

In mathematics, quadratic equations are a fundamental concept, represented by the general form ax² + bx + c = 0. These equations have a rich structure, and their solutions, known as zeroes or roots, are closely related to the coefficients a, b, and c. This article delves into a specific problem where we aim to find the values of the coefficients a and c given information about the zeroes of the equation. Specifically, we'll explore the scenario where the sum and product of the zeroes are equal.

Understanding Quadratic Equations and Zeroes

Before diving into the problem-solving process, it's essential to have a solid grasp of quadratic equations and their zeroes. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. The coefficients a, b, and c are constants, with a not equal to zero (otherwise, it would be a linear equation). The zeroes of a quadratic equation are the values of x that make the equation equal to zero. These are also the points where the graph of the quadratic function (a parabola) intersects the x-axis.

There is a powerful connection between the zeroes of a quadratic equation and its coefficients. If m and n are the zeroes of the quadratic equation ax² + bx + c = 0, then the following relationships hold:

  • Sum of zeroes: m + n = -b/a
  • Product of zeroes: m n = c/a

These formulas are derived from Vieta's formulas and provide a valuable tool for connecting the solutions of a quadratic equation to its coefficients. They allow us to solve various problems, including finding unknown coefficients when information about the zeroes is provided.

The Problem: Finding 'a' and 'c'

Now, let's consider the specific problem we're addressing. We are given the quadratic equation ax² – 12x + c = 0. The problem states that m and n are the zeroes of this equation. Moreover, we are given the crucial piece of information that the sum of the zeroes is equal to the product of the zeroes, and both are equal to 3. In other words, m + n = m n = 3. Our task is to determine the values of the coefficients a and c that satisfy these conditions. This problem highlights how understanding the relationship between roots and coefficients can help us solve for unknowns.

Solving for 'a' and 'c': A Step-by-Step Approach

To solve this problem, we will utilize the relationships between the zeroes and the coefficients that we discussed earlier. We know that for the quadratic equation ax² – 12x + c = 0, the following relationships hold:

  • m + n = -(-12)/a = 12/a
  • m n = c/a

We are also given that m + n = 3 and m n = 3. We can now substitute these values into the equations above to create a system of equations:

  • 3 = 12/a
  • 3 = c/a

This system of equations allows us to solve for the unknowns a and c. Let's start by solving the first equation for a. Multiplying both sides of the equation 3 = 12/a by a, we get:

  • 3a = 12

Now, dividing both sides by 3, we find the value of a:

  • a = 12/3 = 4

We have successfully determined that a = 4. Next, we can substitute this value into the second equation, 3 = c/a, to solve for c. Substituting a = 4, we get:

  • 3 = c/4

Multiplying both sides of this equation by 4, we find the value of c:

  • c = 3 * 4 = 12

Therefore, we have found that c = 12. Thus, we have solved for both a and c based on the given information about the zeroes of the quadratic equation.

Verification: Ensuring Accuracy

To ensure that our solution is correct, it's always a good practice to verify our results. We found that a = 4 and c = 12. Let's substitute these values back into the original quadratic equation:

  • 4x² – 12x + 12 = 0

Now, let's find the zeroes of this equation. To simplify, we can divide the entire equation by 4:

  • x² – 3x + 3 = 0

To find the zeroes, we can use the quadratic formula, which states that for an equation of the form ax² + bx + c = 0, the solutions are given by:

  • x = (-b ± √(b² – 4ac)) / 2a

In our simplified equation, a = 1, b = -3, and c = 3. Substituting these values into the quadratic formula, we get:

  • x = (3 ± √((-3)² – 4 * 1 * 3)) / (2 * 1)
  • x = (3 ± √(9 – 12)) / 2
  • x = (3 ± √(-3)) / 2
  • x = (3 ± i√3) / 2

The zeroes of the equation are complex numbers: (3 + i√3) / 2 and (3 - i√3) / 2. Let's denote these zeroes as m and n, respectively. Now, let's check if m + n = 3 and m n = 3:

  • m + n = (3 + i√3) / 2 + (3 - i√3) / 2 = 6/2 = 3
  • m n = ((3 + i√3) / 2) * ((3 - i√3) / 2) = (9 + 3) / 4 = 12/4 = 3

As we can see, the conditions m + n = 3 and m n = 3 are indeed satisfied. This confirms that our solution, a = 4 and c = 12, is correct.

Implications and Applications

This problem demonstrates the powerful connection between the zeroes and coefficients of a quadratic equation. By understanding and applying the relationships derived from Vieta's formulas, we can solve for unknown coefficients given information about the roots. This skill is valuable in various mathematical contexts and has applications in fields such as physics, engineering, and computer science. For instance, in physics, quadratic equations are used to model projectile motion, and understanding the roots of the equation can help determine the range and maximum height of the projectile.

Furthermore, this problem highlights the importance of verification in mathematical problem-solving. By substituting the values we found back into the original equation and checking if the given conditions are met, we ensure the accuracy of our solution and avoid potential errors. This practice is crucial for developing strong mathematical reasoning and problem-solving skills.

In conclusion, we successfully determined the values of a and c in the quadratic equation ax² – 12x + c = 0, given that m and n are its zeroes and m + n = m n = 3. The values we found are a = 4 and c = 12. This problem illustrates the significance of the relationships between the zeroes and coefficients of a quadratic equation and the importance of verifying our solutions. Understanding these concepts allows us to effectively solve a wide range of problems related to quadratic equations.

Further Exploration

If you found this problem interesting, there are many other aspects of quadratic equations to explore. Some potential areas for further investigation include:

  • The discriminant: The discriminant (b² – 4ac) of a quadratic equation provides valuable information about the nature of its roots (real, complex, distinct, or repeated).
  • Completing the square: This technique can be used to rewrite a quadratic equation in a form that makes it easier to solve.
  • Graphing quadratic functions: Understanding the relationship between the coefficients of a quadratic equation and the graph of the corresponding parabola can provide further insights into the solutions of the equation.

By delving deeper into these topics, you can further enhance your understanding of quadratic equations and their applications in various fields.