Finding The Quadratic Equation With Roots -3 And -7

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In this article, we will explore how to determine the quadratic equation when the roots are given as -3 and -7. This is a fundamental concept in algebra, and understanding it can help solve various mathematical problems. We'll delve into the underlying principles, step-by-step methods, and illustrative examples to ensure a comprehensive grasp of the topic.

Understanding Quadratic Equations and Roots

Before we dive into solving the specific problem, let's establish a solid understanding of quadratic equations and roots. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The roots of a quadratic equation are the values of 'x' that satisfy the equation, meaning they make the equation true when substituted. In simpler terms, the roots are the points where the parabola represented by the quadratic equation intersects the x-axis.

There are several methods to find the roots of a quadratic equation, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a universal method that works for all quadratic equations and is given by:

x = (-b ± √(b² - 4ac)) / 2a

However, in this case, we are given the roots and need to work backward to find the quadratic equation. This involves understanding the relationship between the roots and the coefficients of the quadratic equation.

Relationship Between Roots and Coefficients

The roots of a quadratic equation are closely related to its coefficients. If we denote the roots as α (alpha) and β (beta), then the following relationships hold:

  1. Sum of the roots (α + β) = -b/a
  2. Product of the roots (αβ) = c/a

These relationships are derived from Vieta's formulas and provide a crucial link between the roots and the quadratic equation's structure. We will use these relationships to construct the quadratic equation from the given roots.

Finding the Quadratic Equation

Now, let's apply our understanding to the specific problem: finding the quadratic equation with roots -3 and -7. We are given:

α = -3

β = -7

Our goal is to find the values of 'a', 'b', and 'c' in the general quadratic equation form ax² + bx + c = 0.

Step-by-Step Method

  1. Calculate the sum of the roots:

    α + β = -3 + (-7) = -10

  2. Calculate the product of the roots:

    αβ = (-3) * (-7) = 21

  3. Use the relationships between roots and coefficients:

    We know that:

    • -b/a = -10
    • c/a = 21

  4. Choose a convenient value for 'a':

    For simplicity, let's assume a = 1. This choice will make the calculations easier, and we can always multiply the entire equation by a constant if needed.

  5. Determine 'b' and 'c':

    • Since -b/a = -10 and a = 1, then -b = -10, so b = 10.
    • Since c/a = 21 and a = 1, then c = 21.

  6. Write the quadratic equation:

    Now that we have a = 1, b = 10, and c = 21, we can write the quadratic equation as:

    x² + 10x + 21 = 0

Verification

To ensure our equation is correct, we can verify that -3 and -7 are indeed the roots by substituting them into the equation:

  • For x = -3:

    (-3)² + 10(-3) + 21 = 9 - 30 + 21 = 0

  • For x = -7:

    (-7)² + 10(-7) + 21 = 49 - 70 + 21 = 0

Both values satisfy the equation, confirming that our quadratic equation is correct.

Alternative Method: Using the Factor Theorem

Another way to approach this problem is by using the Factor Theorem. The Factor Theorem states that if 'α' is a root of a polynomial equation, then (x - α) is a factor of the polynomial. Therefore, if -3 and -7 are the roots, then (x + 3) and (x + 7) are factors of the quadratic equation.

Steps using the Factor Theorem

  1. Write the factors:

    Since the roots are -3 and -7, the factors are (x + 3) and (x + 7).

  2. Multiply the factors:

    (x + 3)(x + 7) = x² + 7x + 3x + 21 = x² + 10x + 21

  3. Set the result equal to zero:

    x² + 10x + 21 = 0

This method directly gives us the quadratic equation, which matches the result we obtained earlier. The Factor Theorem provides a more intuitive way to construct the equation when the roots are known.

Examples and Applications

To further solidify our understanding, let's consider a few more examples and applications of finding quadratic equations from given roots.

Example 1

Find the quadratic equation with roots 2 and -5.

  1. Sum of roots: 2 + (-5) = -3
  2. Product of roots: 2 * (-5) = -10
  3. Using the relationships: -b/a = -3, c/a = -10
  4. Let a = 1: b = 3, c = -10
  5. Quadratic equation: x² + 3x - 10 = 0

Example 2

Find the quadratic equation with roots -1/2 and 3.

  1. Sum of roots: -1/2 + 3 = 5/2
  2. Product of roots: (-1/2) * 3 = -3/2
  3. Using the relationships: -b/a = 5/2, c/a = -3/2
  4. Let a = 2: -b = 5, c = -3, so b = -5
  5. Quadratic equation: 2x² - 5x - 3 = 0

These examples illustrate that the method remains consistent regardless of the nature of the roots, whether they are integers, fractions, or negative numbers. The key is to correctly apply the relationships between the roots and coefficients or use the Factor Theorem effectively.

Applications in Real-World Problems

Finding quadratic equations from roots is not just a theoretical exercise; it has practical applications in various fields. For instance, in physics, quadratic equations are used to model projectile motion. If we know the points where a projectile lands (the roots), we can determine the equation describing its trajectory. Similarly, in engineering, quadratic equations are used in designing structures, circuits, and systems. Understanding how to construct these equations from given roots can help in solving design and optimization problems.

Conclusion

In conclusion, finding the quadratic equation with roots -3 and -7 involves understanding the fundamental relationship between the roots and the coefficients of the equation. We explored two primary methods: using the sum and product of roots and applying the Factor Theorem. Both methods lead to the same quadratic equation: x² + 10x + 21 = 0. By mastering this concept, you can solve a wide range of problems in algebra and related fields. Remember to practice with different examples and applications to strengthen your understanding and problem-solving skills. Whether you're a student learning algebra or a professional applying mathematical principles, this knowledge will undoubtedly prove valuable.

The ability to derive a quadratic equation from its roots is a cornerstone of algebraic understanding. This skill not only reinforces the relationships between roots and coefficients but also lays the groundwork for more advanced mathematical concepts. As we've seen, the process involves calculating the sum and product of the roots and then using these values to construct the equation. Alternatively, the Factor Theorem provides a direct pathway by leveraging the roots to form factors and subsequently the quadratic expression. The examples provided, ranging from integer roots to fractional roots, underscore the versatility of these methods. Furthermore, recognizing the real-world applications in fields like physics and engineering highlights the practical relevance of this algebraic technique. By grasping and applying these concepts, individuals can confidently tackle a variety of mathematical challenges and appreciate the interconnectedness of algebraic principles. This article serves as a comprehensive guide, empowering readers to not only solve specific problems but also to deepen their overall understanding of quadratic equations and their applications.