Introduction to Quadratic Equations

Understanding Quadratics

A quadratic is a type of equation that involves a squared term. It’s called "quadratic" because it comes from the Latin word quadratus, meaning "square." The general form of a quadratic equation is: $$ax^2 + bx + c = 0$$

Where:

• $a$ is the coefficient of $x^2$ (it must not be 0),
• $b$ is the coefficient of $x$,
• $c$ is the constant term.

Quadratics create U-shaped curves called parabolas when plotted on a graph. If $a \gt 0$, the parabola opens upwards, and if $a \lt 0$, it opens downwards.

 

 

Additionally, quadratic equations can have either two real solutions, one real solution (when the curve just touches the x-axis), or no real solutions (when the curve does not intersect the x-axis).

 

Key Features of Quadratics

1. The Highest Power of $x$ is 2: The squared term, $x^2$, is what makes the equation quadratic. Example: $$x^2 + 5x + 6 \text{is quadratic (highest power is} x^2 \text{)} \\ 3x + 7 \text{is linear (no} x^2 \text{term)}$$

2. The Shape on a Graph: Quadratics always make parabolas. For example: $$y = x^2 \text{is a simple U-shape} \\ y = -x^2 \text{is an upside-down U-shape}$$
3. Real-Life Uses: Quadratics show up in physics (e.g., projectile motion), economics (e.g., profit curves), and many more fields

 

Solving Quadratic Equations

There are several methods for solving quadratic equations, each with its own application and advantage:

• Factorising: Involves expressing the quadratic equation as a product of two binomial expressions. This method works well for quadratics that can be easily factorised.
• Completing the Square: Entails manipulating the equation to create a perfect square trinomial, making it easier to solve. This method is useful for deriving the Quadratic Formula and solving equations that are not easily factorised.
• The Quadratic Formula: A universal solution method derived from completing the square, given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can solve any quadratic equation, including those that cannot be factorised.