2. Methods of Solving Quadratic Equations

2.1 Factorization Method

Step-by-step:

1. Write the equation in standard form.
2. Factor the quadratic expression on the LHS.
3. Set each factor equal to 0.
4. Solve for 𝑥.

Example: Solve 𝑥² + 5𝑥 + 6 = 0 Solution: 𝑥² + 5𝑥 + 6 = (𝑥 + 2)(𝑥 + 3) = 0 So, 𝑥 = -2 or 𝑥 = -3

The Factorization Method involves breaking down the quadratic equation into simpler binomial expressions. First, the equation must be presented in the standard form (𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0). Next, we identify two numbers that multiply to 'c' (the constant term) and add to 'b' (the coefficient of the linear term). Once we identify these numbers, we express the quadratic as the product of two binomials. Finally, we set each binomial equal to zero to find the possible values of 𝑥.

For example, in the equation 𝑥² + 5𝑥 + 6 = 0, we can factor it into (𝑥 + 2)(𝑥 + 3) = 0. Solving for each factor gives us the roots: 𝑥 = -2 or 𝑥 = -3.

Imagine you're trying to find two numbers that, when combined, will create a certain length of a fence (the constant) and also balance out evenly on a number line (the coefficient). This is similar to finding the right pairs of factors that multiply together to meet the requirements of the quadratic equation.

2.2 Completing the Square Method

Step-by-step:

1. Rearrange the equation: move the constant to the other side.
2. Make the coefficient of 𝑥² equal to 1 (if necessary).
3. Add and subtract the square of half the coefficient of 𝑥.
4. Rewrite as a perfect square and solve.

Example: Solve 𝑥² + 6𝑥 + 5 = 0 by completing the square. Solution: 𝑥² + 6𝑥 = -5 Add 9 to both sides: 𝑥² + 6𝑥 + 9 = 4 (𝑥 + 3)² = 4 𝑥 + 3 = ±2 𝑥 = -1 or 𝑥 = -5

The Completing the Square Method is a process for transforming a quadratic equation into a perfect square trinomial. First, move the constant term to the opposite side of the equation. If the coefficient of 𝑥² is not 1, divide through by that coefficient to make it 1. Next, take half of the coefficient of 𝑥, square it, and add it to both sides of the equation. This allows us to rewrite the left side as a perfect square. Finally, we solve for 𝑥 by taking the square root of both sides and isolating 𝑥.

In our example, starting with 𝑥² + 6𝑥 + 5 = 0, we rearranged to 𝑥² + 6𝑥 = -5, added 9 to both sides to make it (𝑥 + 3)² = 4, leading us to find 𝑥 = -1 or 𝑥 = -5.

Think of it like adjusting a picture frame until it fits perfectly within a space. By adding or removing material (like adjusting the coefficients), you can make everything balance, revealing a clean and complete picture—just as completing the square helps you visualize the roots of the quadratic equation.

2.3 Quadratic Formula

The quadratic formula: −𝑏 ± √(𝑏² - 4𝑎𝑐)
𝑥 =
2𝑎 Works for any quadratic equation.

Example: Solve 2𝑥² + 3𝑥 − 2 = 0 Solution: 𝑎 = 2, 𝑏 = 3, 𝑐 = −2 −3 ± √(9 + 16)
−3 ± √(25)
𝑥 =
2
4 𝑥 = 0.5 or 𝑥 = −2

The Quadratic Formula is a universal method that can solve any quadratic equation in the standard form (𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0). It is given by the formula: 𝑥 = [−𝑏 ± √(𝑏² - 4𝑎𝑐)] / 2𝑎. Here, 𝑏² - 4𝑎𝑐 is known as the discriminant. The nature of the roots can be inferred from the discriminant, and it will always provide solutions to quadratic equations, even when the roots are complex.

For instance, for the equation 2𝑥² + 3𝑥 − 2 = 0, substituting the values of 𝑎, 𝑏, and 𝑐 into the quadratic formula yields the solutions 𝑥 = 0.5 and 𝑥 = -2.

Imagine trying to find the optimal path while climbing a hill: the quadratic formula is like your GPS—it will direct you toward your destination (the roots) no matter the terrain (the specific coefficients). Just as the GPS provides accurate guidance, the quadratic formula gives you the exact values for 𝑥.

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