Solving Quadratic Equations Step-by-Step Guide

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Quadratic equations are fundamental in algebra, appearing in various mathematical and real-world applications. Mastering the techniques to solve them is essential for any student or professional dealing with mathematical problems. This article provides a detailed walkthrough of solving several quadratic equations, complete with explanations and hints to ensure a clear understanding. We will cover three distinct equations, addressing different methods and challenges that may arise.

H2: 1. Solving 3a²x² + 8abx + 4b² = 0, a ≠ 0

Unpacking the Quadratic Equation

Our journey begins with the quadratic equation 3a²x² + 8abx + 4b² = 0, where a ≠ 0. The condition a ≠ 0 is crucial because it ensures that the equation remains quadratic; if a were zero, the x² term would vanish, and the equation would become linear. To tackle this equation, we will employ the method of factoring, a technique that involves breaking down the quadratic expression into two binomial factors. Factoring is an elegant and efficient method when applicable, providing direct insight into the solutions or roots of the equation.

Strategic Factoring Techniques

The initial step in factoring involves identifying two binomials that, when multiplied, yield the original quadratic expression. We look for two expressions of the form (Ax + B) and (Cx + D) such that (Ax + B)(Cx + D) = 3a²x² + 8abx + 4b². To find these binomials, we need to consider the coefficients of the terms in the quadratic equation. The product of the first terms in the binomials (A * C) must equal the coefficient of x², which is 3a². Similarly, the product of the last terms (B * D) must equal the constant term, 4b². The sum of the cross products (ADx + BCx) must equal the middle term, 8abx.

Detailed Step-by-Step Solution

Let's systematically break down the factoring process:

  1. Identify Possible Factors: For 3a²x², the factors could be (3ax) and (ax). For 4b², the factors could be (2b) and (2b) or (4b) and (b). We need to find the combination that satisfies the middle term, 8abx.
  2. Trial and Error: We try different combinations. If we choose (3ax + 2b) and (ax + 2b), multiplying these gives 3a²x² + 6abx + 2abx + 4b² = 3a²x² + 8abx + 4b². This combination works perfectly!
  3. Write the Factored Form: So, the factored form of the quadratic equation is (3ax + 2b)(ax + 2b) = 0.
  4. Set Each Factor to Zero: To find the solutions for x, we set each factor equal to zero:
    • 3ax + 2b = 0
    • ax + 2b = 0
  5. Solve for x: Solving these equations for x gives us:
    • From 3ax + 2b = 0, we get 3ax = -2b, so x = -2b / (3a).
    • From ax + 2b = 0, we get ax = -2b, so x = -2b / a.

Concluding the Solution

Therefore, the solutions for the quadratic equation 3a²x² + 8abx + 4b² = 0 are x = -2b / (3a) and x = -2b / a. These values of x are the roots of the equation, and they represent the points where the quadratic function intersects the x-axis. Factoring, as demonstrated here, is a powerful tool for solving quadratic equations, allowing us to efficiently find these roots by breaking down the equation into simpler components.

H2: 2. Solving 4x² - 4ax + (a² - b²) = 0, where a, b ∈ ℝ

Introduction to the Problem

Next, we tackle the equation 4x² - 4ax + (a² - b²) = 0, where a and b are real numbers (a, b ∈ ℝ). This equation presents an interesting challenge because it involves both a and b as parameters, which affect the solutions for x. To solve this, we can employ several methods, including factoring, completing the square, or using the quadratic formula. The hint provided suggests a factoring approach by rewriting the middle term, which we will explore in detail.

Factoring Strategy

The suggested approach involves rewriting the -4ax term as -2(a + b)x - 2(a - b)x. This decomposition is strategic because it sets the stage for factoring by grouping, a technique that allows us to break down the quadratic expression into more manageable parts. The key to successful factoring by grouping is to identify common factors within these groups, which can then be extracted to simplify the expression.

Step-by-Step Factoring Process

Let's follow the factoring process step by step:

  1. Rewrite the Middle Term: Rewrite the equation using the hint:

    4x² - 2(a + b)x - 2(a - b)x + (a² - b²) = 0

  2. Group Terms: Group the first two terms and the last two terms together:

    (4x² - 2(a + b)x) + (-2(a - b)x + (a² - b²)) = 0

  3. Factor out Common Factors: Factor out the greatest common factor from each group:

    • From the first group, factor out 2x: 2x(2x - (a + b))
    • From the second group, factor out -(a - b): -(a - b)(2x - (a + b))

  4. Combine Factors: Now, the equation looks like this:

    2x(2x - (a + b)) - (a - b)(2x - (a + b)) = 0

    Notice that (2x - (a + b)) is a common factor. Factor it out:

    (2x - (a + b))(2x - (a - b)) = 0

Solving for x

Now that we have factored the quadratic equation, we can find the solutions for x by setting each factor equal to zero:

  1. Set the First Factor to Zero: 2x - (a + b) = 0
    • Solve for x: 2x = a + b, so x = (a + b) / 2
  2. Set the Second Factor to Zero: 2x - (a - b) = 0
    • Solve for x: 2x = a - b, so x = (a - b) / 2

Final Solutions

The solutions for the quadratic equation 4x² - 4ax + (a² - b²) = 0 are x = (a + b) / 2 and x = (a - b) / 2. These solutions depend on the values of a and b, highlighting how parameters in a quadratic equation can influence its roots. This factoring method, facilitated by the strategic rewriting of the middle term, provides an elegant way to solve this equation.

H2: 3. Solving 5x² - 12x - 9 = 0 for x ∈ ℤ and x ∈ ℚ

Introduction to the Problem

Lastly, we consider the quadratic equation 5x² - 12x - 9 = 0. This equation presents an interesting twist because we are asked to find solutions within specific number sets: integers (ℤ) and rational numbers (ℚ). This constraint means we need to not only solve the equation but also check whether the solutions meet these criteria. Factoring and the quadratic formula are both viable methods here, and we will explore them to find the solutions.

Exploring Solution Sets: Integers and Rationals

Before diving into the solution process, it’s crucial to understand the distinction between integers and rational numbers. Integers are whole numbers (positive, negative, or zero), while rational numbers are numbers that can be expressed as a fraction p / q, where p and q are integers and q ≠ 0. All integers are rational numbers (since any integer n can be written as n / 1), but not all rational numbers are integers (e.g., 1 / 2 is rational but not an integer).

Method 1: Factoring

Let's attempt to solve 5x² - 12x - 9 = 0 by factoring:

  1. Look for Factors: We need to find two binomials (Ax + B) and (Cx + D) such that (Ax + B)(Cx + D) = 5x² - 12x - 9. The product of the first terms (A * C) should be 5, and the product of the last terms (B * D) should be -9. The sum of the cross products (ADx + BCx) should be -12x.
  2. Trial and Error: Possible factors for 5x² are (5x) and (x). Possible factors for -9 are (3) and (-3) or (9) and (-1). After some trial and error, we find that (5x + 3)(x - 3) = 5x² - 15x + 3x - 9 = 5x² - 12x - 9. This factorization works!
  3. Write the Factored Form: So, the factored form of the quadratic equation is (5x + 3)(x - 3) = 0.
  4. Set Each Factor to Zero: To find the solutions for x, we set each factor equal to zero:
    • 5x + 3 = 0
    • x - 3 = 0
  5. Solve for x: Solving these equations for x gives us:
    • From 5x + 3 = 0, we get 5x = -3, so x = -3 / 5.
    • From x - 3 = 0, we get x = 3.

Method 2: Quadratic Formula

Alternatively, we can use the quadratic formula to solve 5x² - 12x - 9 = 0. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In this case, a = 5, b = -12, and c = -9.

  1. Plug in the Values: Substitute the values into the formula:

    x = (12 ± √((-12)² - 4 * 5 * (-9))) / (2 * 5)

  2. Simplify: Simplify the expression:

    x = (12 ± √(144 + 180)) / 10

    x = (12 ± √324) / 10

    x = (12 ± 18) / 10

  3. Find the Solutions: The two solutions are:

    • x = (12 + 18) / 10 = 30 / 10 = 3
    • x = (12 - 18) / 10 = -6 / 10 = -3 / 5

Identifying Solutions within Number Sets

Now, let's consider the solutions we found: x = 3 and x = -3 / 5.

  1. For x ∈ ℤ: The integer solutions are those that are whole numbers. In this case, x = 3 is an integer.
  2. For x ∈ ℚ: The rational solutions are those that can be expressed as a fraction. Both x = 3 (which can be written as 3 / 1) and x = -3 / 5 are rational numbers.

Concluding the Solution

For the quadratic equation 5x² - 12x - 9 = 0, the solutions are:

  • When x ∈ ℤ, the solution is x = 3.
  • When x ∈ ℚ, the solutions are x = 3 and x = -3 / 5.

This example highlights the importance of considering the specified number sets when solving equations, as it can restrict the set of valid solutions.

In conclusion, solving quadratic equations requires a versatile toolkit of methods, including factoring, completing the square, and using the quadratic formula. Each method has its strengths, and the choice of method often depends on the specific equation and the constraints of the problem. By understanding these techniques and practicing their application, one can confidently tackle a wide range of quadratic equations and related problems. This article has walked through three distinct examples, each illustrating different aspects of solving quadratic equations, providing a solid foundation for further exploration in algebra and mathematics.