Solving Quadratic Equations

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What is a Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x with the general form:

Ax² + Bx + C = 0

Where A, B, and C are constants, and x represents an unknown variable. The highest exponent of x is 2.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, including factoring, completing the square, using the quadratic formula, and graphing. Each method is useful in different scenarios.

1. Factoring

Factoring involves rewriting the quadratic equation as a product of two binomials set to zero.

Example:

x² - 5x + 6 = 0

Step 1: Factor the quadratic expression:

(x - 2)(x - 3) = 0

Step 2: Set each factor to zero:

x - 2 = 0 or x - 3 = 0

Step 3: Solve for x:

x = 2 or x = 3

2. Completing the Square

Completing the square transforms the quadratic equation into a perfect square trinomial, making it easier to solve.

Example:

x² + 6x + 5 = 0

Step 1: Move the constant term to the other side:

x² + 6x = -5

Step 2: Add and subtract the square of half the coefficient of x:

x² + 6x + 9 = -5 + 9

Step 3: Rewrite as a perfect square trinomial:

(x + 3)² = 4

Step 4: Take the square root of both sides:

x + 3 = ±2

Step 5: Solve for x:

x = -1 or x = -5

3. Quadratic Formula

The quadratic formula provides a straightforward way to find the roots of any quadratic equation:


                x =
                
                    -B ± √(B² - 4AC)
                    2A

Use this formula when factoring is difficult or impossible.

Example:

2x² + 4x - 6 = 0

Step 1: Identify A, B, and C:

A = 2, B = 4, C = -6

Step 2: Plug these values into the quadratic formula:


                x =
                
                    -4 ± √(4² - 4 × 2 × (-6))
                    2 × 2

Step 3: Simplify inside the square root:


                    x =
                    
                        -4 ± √(16 + 48)
                        4

Step 4: Simplify further:

, then

, and then you get

Step 5: Solve for x:

x = 1 or x = -3

4. Graphing

Graphing involves plotting the quadratic equation on a coordinate plane and identifying the points where it crosses the x-axis (roots).

Example:

y = x² - 4x + 3

Step 1: Plot the vertex and a few points on either side of the vertex.

Step 2: Identify the points where the graph intersects the x-axis.

Result: x = 1 and x = 3

Key Points to Remember

Factoring works well for simple quadratics where factors are easily identifiable.
Completing the square is useful for transforming any quadratic equation into a form that can be solved by taking the square root.
The quadratic formula is a universal method that works for all quadratic equations.
Graphing provides a visual representation of the roots and the behavior of the quadratic function.
Always check your solutions by plugging them back into the original equation.