Solve The Quadratic Equation 8x² - 22x - 21 = 0 By Factorization.

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In the realm of mathematics, quadratic equations hold a significant position, arising in various fields, from physics and engineering to economics and computer science. Among the methods to solve these equations, factorization stands out as a fundamental and insightful technique. This article delves into the process of solving the quadratic equation 8x² - 22x - 21 = 0 by factorization, providing a step-by-step guide and exploring the underlying principles.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of x that satisfy the equation.

Solving quadratic equations involves finding these roots, which can be achieved through various methods, including factorization, completing the square, using the quadratic formula, and graphical methods. Factorization is a powerful technique that relies on expressing the quadratic expression as a product of two linear factors. When the product of these factors is zero, at least one of the factors must be zero, leading to the solutions of the equation.

Factorization Method: A Step-by-Step Approach

Factorization involves rewriting the quadratic equation in the form (px + q)(rx + s) = 0, where p, q, r, and s are constants. To achieve this, we need to find two numbers that multiply to give the product of the leading coefficient (a) and the constant term (c), and add up to the coefficient of the linear term (b). This process can be broken down into the following steps:

  1. Identify the coefficients: In the given equation, 8x² - 22x - 21 = 0, we have a = 8, b = -22, and c = -21.
  2. Calculate the product of a and c: ac = 8 * (-21) = -168.
  3. Find two numbers: Now, we need to find two numbers that multiply to -168 and add up to -22. Through trial and error or by considering the factors of -168, we find that the numbers -28 and 6 satisfy these conditions (-28 * 6 = -168 and -28 + 6 = -22).
  4. Rewrite the middle term: Replace the middle term (-22x) with the sum of the two terms formed using the numbers we found (-28x + 6x). The equation becomes 8x² - 28x + 6x - 21 = 0.
  5. Factor by grouping: Group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group. From the first group (8x² - 28x), the GCF is 4x, and from the second group (6x - 21), the GCF is 3. This gives us 4x(2x - 7) + 3(2x - 7) = 0.
  6. Factor out the common binomial: Notice that both terms now have a common binomial factor (2x - 7). Factor this out to get (2x - 7)(4x + 3) = 0.
  7. Set each factor to zero: Since the product of the two factors is zero, at least one of them must be zero. Set each factor equal to zero and solve for x:
    • 2x - 7 = 0 => 2x = 7 => x = 7/2
    • 4x + 3 = 0 => 4x = -3 => x = -3/4

Therefore, the solutions to the quadratic equation 8x² - 22x - 21 = 0 are x = 7/2 and x = -3/4.

Alternative Methods for Solving Quadratic Equations

While factorization is a powerful technique, it's not always the most efficient method, especially when the factors are not readily apparent or when the equation has irrational roots. In such cases, other methods like the quadratic formula or completing the square might be more suitable.

Quadratic Formula

The quadratic formula is a general formula that provides the solutions to any quadratic equation of the form ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

Applying this formula to the equation 8x² - 22x - 21 = 0, we get:

x = (22 ± √((-22)² - 4 * 8 * (-21))) / (2 * 8) x = (22 ± √(484 + 672)) / 16 x = (22 ± √1156) / 16 x = (22 ± 34) / 16

This gives us two solutions:

x = (22 + 34) / 16 = 56 / 16 = 7/2 x = (22 - 34) / 16 = -12 / 16 = -3/4

As we can see, the solutions obtained using the quadratic formula are the same as those obtained by factorization.

Completing the Square

Completing the square is another method for solving quadratic equations that involves rewriting the equation in the form (x + p)² = q, where p and q are constants. This method is particularly useful when the quadratic equation cannot be easily factored. While we won't delve into the details of completing the square in this article, it's worth noting that it's a valuable technique to have in your mathematical toolkit.

Importance of Factorization

Factorization is not just a method for solving quadratic equations; it's a fundamental skill in algebra and calculus. It's used in simplifying expressions, solving higher-degree equations, and in various applications in calculus and other areas of mathematics. Mastering factorization provides a strong foundation for understanding more advanced mathematical concepts.

Common Mistakes to Avoid

When factorizing quadratic equations, it's essential to avoid common mistakes that can lead to incorrect solutions. Some of these mistakes include:

  • Incorrectly identifying the factors of the constant term.
  • Making errors in arithmetic when adding or multiplying the factors.
  • Forgetting to set each factor to zero after factorization.
  • Failing to simplify the solutions obtained.

By being mindful of these potential pitfalls, you can improve your accuracy and confidence in solving quadratic equations by factorization.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used to model projectile motion, calculate areas and volumes, optimize processes in engineering and economics, and in many other fields. Understanding how to solve quadratic equations is crucial for anyone pursuing a career in science, technology, engineering, or mathematics.

Conclusion

Solving quadratic equations by factorization is a fundamental skill in mathematics. This article has provided a detailed, step-by-step guide to solving the equation 8x² - 22x - 21 = 0 using factorization. We have also explored alternative methods like the quadratic formula and discussed the importance of factorization in mathematics and its real-world applications. By mastering this technique, you can enhance your problem-solving abilities and gain a deeper understanding of mathematical principles.

Remember, practice is key to mastering any mathematical skill. Work through various examples and challenge yourself with different types of quadratic equations to become proficient in factorization and other methods for solving quadratic equations.

In summary, solving the quadratic equation 8x² - 22x - 21 = 0 by factorization involves finding two numbers that multiply to -168 (the product of 8 and -21) and add up to -22. These numbers are -28 and 6. Rewriting the equation and factoring by grouping leads to the factors (2x - 7) and (4x + 3), which when set to zero, give the solutions x = 7/2 and x = -3/4. This method, along with the quadratic formula and completing the square, provides a robust toolkit for tackling quadratic equations in various contexts.