Equation With Solutions M=-5 And M=9

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Understanding how to construct a quadratic equation given its solutions is a fundamental concept in algebra. This article will guide you through the process of determining the equation that has solutions m = -5 and m = 9. We'll explore the underlying principles, walk through the steps, and provide a clear explanation to help you grasp this important mathematical skill.

Introduction to Quadratic Equations and Solutions

Quadratic equations are polynomial equations of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The solutions of a quadratic equation, also known as roots or zeros, are the values of the variable that satisfy the equation. In other words, when you substitute these values back into the equation, the equation holds true.

One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. For example, the quadratic expression x² + 5x + 6 can be factored as (x + 2)(x + 3). The solutions to the equation x² + 5x + 6 = 0 are then the values of x that make each factor equal to zero, which are x = -2 and x = -3.

The relationship between the solutions and the factors of a quadratic equation is crucial for understanding how to construct an equation given its solutions. If m = r is a solution to a quadratic equation, then (m - r) must be a factor of the quadratic expression. This is because when m = r, the factor (m - r) becomes zero, making the entire expression equal to zero.

Constructing the Equation from Solutions

To construct a quadratic equation with the solutions m = -5 and m = 9, we need to reverse the factoring process. We start by identifying the factors that correspond to each solution. Since m = -5 is a solution, the factor (m - (-5)) or (m + 5) must be present. Similarly, since m = 9 is a solution, the factor (m - 9) must be present.

Therefore, the quadratic equation can be written as the product of these factors set equal to zero:

(m + 5)(m - 9) = 0

This equation represents the factored form of the quadratic equation. To obtain the standard form (am² + bm + c = 0), we need to expand the product of the factors. This involves using the distributive property (also known as the FOIL method) to multiply each term in the first factor by each term in the second factor:

(m + 5)(m - 9) = m(m - 9) + 5(m - 9)

= m² - 9m + 5m - 45

= m² - 4m - 45

Thus, the quadratic equation in standard form is:

m² - 4m - 45 = 0

This equation has the solutions m = -5 and m = 9. We can verify this by substituting these values back into the equation and confirming that the equation holds true.

Step-by-Step Solution

Let's outline the steps to find the equation with solutions m = -5 and m = 9:

  1. Identify the solutions: We are given the solutions m = -5 and m = 9.
  2. Form the factors: For each solution m = r, the corresponding factor is (m - r). Therefore, the factors are (m - (-5)) = (m + 5) and (m - 9).
  3. Write the factored form of the equation: Multiply the factors and set the product equal to zero: (m + 5)(m - 9) = 0.
  4. Expand the factors (optional): To obtain the standard form, expand the product of the factors: (m + 5)(m - 9) = m² - 9m + 5m - 45 = m² - 4m - 45.
  5. Write the equation in standard form (optional): Set the expanded expression equal to zero: m² - 4m - 45 = 0.

Following these steps allows us to confidently construct the quadratic equation with the given solutions.

Analyzing the Given Options

Now, let's analyze the options provided in the problem statement to determine which one has the solutions m = -5 and m = 9:

  • (m + 5)(m - 9) = 0: This option directly represents the factored form of the equation with the correct factors corresponding to the solutions m = -5 and m = 9. Therefore, this is the correct equation.
  • (m - 5)(m + 9) = 0: This option has factors that correspond to the solutions m = 5 and m = -9, which are not the solutions we are looking for. So, this option is incorrect.
  • m² - 5m + 9 = 0: This option is in standard form, but it does not have the correct solutions. We can verify this by using the quadratic formula or by attempting to factor the expression. It will not yield the solutions m = -5 and m = 9. Therefore, this option is incorrect.
  • m² + 5m - 9 = 0: This option is also in standard form and does not have the correct solutions. Similar to the previous option, using the quadratic formula or factoring will not result in the solutions m = -5 and m = 9. Therefore, this option is incorrect.

Based on our analysis, the only equation that has solutions m = -5 and m = 9 is (m + 5)(m - 9) = 0.

Importance of Understanding the Relationship Between Solutions and Factors

The ability to construct an equation from its solutions is a vital skill in algebra. It demonstrates a deep understanding of the relationship between solutions, factors, and the quadratic equation itself. This understanding is essential for solving various mathematical problems, including:

  • Curve sketching: Knowing the roots of a quadratic equation helps determine the points where the parabola intersects the x-axis, which is crucial for sketching the graph.
  • Optimization problems: In many optimization problems, finding the roots of a quadratic equation helps identify the maximum or minimum values of a function.
  • Modeling real-world scenarios: Quadratic equations are used to model various real-world phenomena, such as projectile motion and the trajectory of objects. Understanding the solutions helps interpret the model and make predictions.
  • Solving higher-degree polynomials: The principles used to construct quadratic equations can be extended to higher-degree polynomials as well.

By mastering this concept, you'll be well-equipped to tackle a wide range of algebraic problems and applications.

Common Mistakes to Avoid

When constructing an equation from its solutions, it's essential to avoid common mistakes that can lead to incorrect results. Here are some pitfalls to watch out for:

  • Incorrectly forming the factors: Remember that if m = r is a solution, the factor is (m - r), not (m + r). For example, if m = -5 is a solution, the factor is (m - (-5)) = (m + 5), not (m - 5). This is a frequent source of error, so pay close attention to the sign.
  • Forgetting to set the product of factors equal to zero: The equation must be in the form (m - r₁)(m - r₂) = 0, where r₁ and r₂ are the solutions. Setting the product equal to zero is crucial for finding the solutions.
  • Errors in expanding the factors: When expanding the product of the factors to obtain the standard form, be careful to apply the distributive property correctly and avoid sign errors. Double-check your work to ensure accuracy.
  • Confusing solutions with coefficients: The solutions of a quadratic equation are not the same as the coefficients in the standard form. The solutions are the values of the variable that make the equation true, while the coefficients are the constants that multiply the variable terms.

By being aware of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence in constructing equations from their solutions.

Practice Problems

To solidify your understanding of constructing equations from solutions, let's work through a few practice problems:

  1. Find the equation with solutions m = 2 and m = -3.
  2. Find the equation with solutions m = -1 and m = 4.
  3. Find the equation with solutions m = -2 and m = -5.

Solutions:

  1. Factors: (m - 2) and (m + 3). Equation: (m - 2)(m + 3) = 0. Expanding: m² + m - 6 = 0.
  2. Factors: (m + 1) and (m - 4). Equation: (m + 1)(m - 4) = 0. Expanding: m² - 3m - 4 = 0.
  3. Factors: (m + 2) and (m + 5). Equation: (m + 2)(m + 5) = 0. Expanding: m² + 7m + 10 = 0.

Working through these practice problems will help you develop your skills and build confidence in solving similar problems.

Conclusion

In conclusion, determining the equation with solutions m = -5 and m = 9 involves understanding the relationship between solutions and factors of a quadratic equation. By forming the factors (m + 5) and (m - 9) and setting their product equal to zero, we arrive at the equation (m + 5)(m - 9) = 0. This concept is fundamental in algebra and has applications in various mathematical and real-world scenarios. Mastering this skill will empower you to solve a wide range of problems involving quadratic equations and their solutions.