Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the form:

$$ ax^2 + bx + c = 0 $$

where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The degree of the equation is 2, which means the highest power of \( x \) is 2.

Quadratic equations are important because they appear in many real-world situations, such as physics, engineering, and economics. The solutions to a quadratic equation can be found using various methods, including factoring, completing the square, and the quadratic formula.

Standard Form of a Quadratic Equation

The most common form of a quadratic equation is called the standard form:

$$ ax^2 + bx + c = 0 $$

In this equation:

  • \( a \) is the coefficient of \( x^2 \),
  • \( b \) is the coefficient of \( x \), and
  • \( c \) is the constant term.

The values of \( a \), \( b \), and \( c \) determine the shape and position of the parabola when the quadratic equation is graphed. The graph of a quadratic equation is always a parabola.

Example 1: Basic Quadratic Equation

Consider the following quadratic equation:

$$ x^2 + 5x + 6 = 0 $$

In this equation:

  • \( a = 1 \),
  • \( b = 5 \),
  • \( c = 6 \).

We can solve this equation using various methods such as factoring, completing the square, or applying the quadratic formula. For now, we'll focus on factoring.

Example 2: Factoring a Quadratic Equation

Let’s factor the equation \( x^2 + 5x + 6 = 0 \).

First, we look for two numbers that multiply to give \( c \) (6) and add to give \( b \) (5). These numbers are 2 and 3, because:

  • 2 × 3 = 6
  • 2 + 3 = 5

Now, we can factor the quadratic equation as follows:

$$ (x + 2)(x + 3) = 0 $$

To solve this, we set each factor equal to zero:

  • \( x + 2 = 0 \Rightarrow x = -2 \)
  • \( x + 3 = 0 \Rightarrow x = -3 \)

The solutions to the quadratic equation are \( x = -2 \) and \( x = -3 \).

The Discriminant and the Nature of Solutions

The discriminant of a quadratic equation is given by the formula:

$$ \Delta = b^2 - 4ac $$

The discriminant helps us determine the nature of the solutions:

  • If \( \Delta > 0 \), there are two distinct real solutions.
  • If \( \Delta = 0 \), there is exactly one real solution (a repeated root).
  • If \( \Delta < 0 \), there are no real solutions (the solutions are complex).

Example 3: Analyzing the Discriminant

Consider the equation:

$$ 2x^2 - 4x + 3 = 0 $$

In this case, \( a = 2 \), \( b = -4 \), and \( c = 3 \). Let’s calculate the discriminant:

$$ \Delta = (-4)^2 - 4(2)(3) = 16 - 24 = -8 $$

Since \( \Delta = -8 \), which is less than 0, there are no real solutions to this equation. The solutions are complex (not real numbers).

Practice Questions

  1. Solve the equation \( x^2 + 6x + 5 = 0 \).
    Solution

    Identify \( a = 1 \), \( b = 6 \), and \( c = 5 \).

    Using factoring:

    $$ x^2 + 6x + 5 = 0 $$

    We find two numbers that multiply to give 5 and add to give 6: 1 and 5.

    $$ (x + 1)(x + 5) = 0 $$

    • \( x = -1 \)
    • \( x = -5 \)

    The solutions are \( x = -1 \) and \( x = -5 \).

  2. Solve the equation \( 3x^2 - 12x + 9 = 0 \).
    Solution

    Identify \( a = 3 \), \( b = -12 \), and \( c = 9 \).

    We can factor this equation:

    $$ 3(x^2 - 4x + 3) = 0 $$

    Factor the quadratic inside the parentheses:

    $$ 3(x - 1)(x - 3) = 0 $$

    • \( x = 1 \)
    • \( x = 3 \)

    The solutions are \( x = 1 \) and \( x = 3 \).

  3. Solve the equation \( x^2 - 4x + 4 = 0 \).
    Solution

    Identify \( a = 1 \), \( b = -4 \), and \( c = 4 \).

    Factor the equation:

    $$ (x - 2)(x - 2) = 0 $$

    • \( x = 2 \) (a repeated root)

    The solution is \( x = 2 \) (a repeated root).

→Quadratic Functions and Their Properties