What Is a Quadratic Equation? (Mathematics Lesson)
What Is a Quadratic Equation?
A quadratic equation is an equation in the form:
On a graph, a quadratic equation looks like a curve:
- a, b and c are constants, they stand in for numbers.
Understanding Quadratic Equations
Let's look at a quadratic equation:
It is the x2 that makes it a quadratic equation.
A quadratic equation is an equation where the highest power of x is 2.
We see x in two places:
x appears with a coefficient of 2 in front of it (2x).
x appears as x2. This is x with an exponent of 2.
Real Examples of Quadratic Equations
Some examples of quadratic equations are shown below. The values of the constants a, b and c are written next to them:
Solving Quadratic EquationsSolving a quadratic equation means finding the value of x that make y = 0:
The values of x are the roots of the quadratic equation.
There are 3 main ways of solving quadratic equations:
- Factor the quadratic equation:
Either bracket, (x - x1) and (x - x2), when equal to 0, makes their product 0.
If (x - x1) = 0 and (x - x2) = 0, then the roots of the equation are:
- Use the quadratic formula:
The ± sign means there are two roots:
- Use the graphical method:
The roots, x1 and x2 are where the quadratic curve crosses the x-axis.
Read more about solving quadratic equations by factoring.
Read more about solving quadratic equations using the quadratic formula.
Read more about solving quadratic equations using the graphical method.
NoteWHAT'S IN A NAME?
The word 'quadratic' comes from the word 'quad', meaning "square" - because the x is squared.
DIFFERENT FORMS OF QUADRATIC EQUATIONSThe many different forms of linear equations may be quite confusing. But they all have some things in common:
- There are 2 variables, y and x.
- There are constants, like 2 or c.
- The highest power of x is 2.
If the highest power of x is 1, (i.e.
a = 0 in the standard form), it is a linear equation:
These are quadratic equations, not linear equations.
Neither will you see powers of 3 (cubic equations) or higher:
The y-intercept is found by inserting x = 0 into the quadratic equation. This gives:
THERE ARE 2 ROOTSQuadratic equations always have two solutions.
Sometimes it seems that there is only one root, but it is repeated.
Even when it seems there are no roots, there are two complex roots.