Welcome to the world of quadratic equations! You've likely solved linear equations like $x+3=5$, but quadratics are the next step up. They are incredibly useful in fields like physics, engineering, and finance.
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually $x$) is 2. The standard form of a quadratic equation is:
$$ax^2 + bx + c = 0$$Here's what the letters mean:
When you plot a quadratic function like $y = ax^2 + bx + c$, it creates a beautiful, symmetrical U-shaped curve called a parabola.
The coefficient '$a$' tells us which way the parabola opens:
When we "solve" the equation $ax^2 + bx + c = 0$, we are finding the roots, which are the x-values where the graph crosses the x-axis (i.e., where $y=0$). A parabola can cross the x-axis twice, once, or not at all.
One of the most common ways to solve a quadratic equation is by factorising. Let's walk through an example.
Step 1: The Goal
We are looking for two numbers that do two things:
Step 2: Find the Numbers
Let's think of pairs of numbers that multiply to make 6:
(1 and 6), (2 and 3), (-1 and -6), (-2 and -3).
Which pair also adds up to 5? It's 2 and 3, since $2+3=5$. We found our numbers!
Step 3: Factorise the Equation
Now we can rewrite the equation using these numbers in two brackets:
Step 4: Find the Solutions
If two things multiply together to make zero, one of them must be zero. This is a key rule in algebra. So, we can say that either the first bracket is zero, or the second bracket is zero.
Either $x+2 = 0 \implies x = -2$
Or $x+3 = 0 \implies x = -3$
The solution is $x = -2$ or $x = -3$. These are the two points on the x-axis where the parabola for $y = x^2 + 5x + 6$ would cross.
Factorising is a great method, but it only works for some quadratics. For others, you'll learn about other techniques like "completing the square" and the "quadratic formula".