What Is a Quadratic Equation?

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2, written in the standard form $a x^{2} + b x + c = 0$ where $a$ , $b$ , and $c$ are constants and $a n e q 0$ .

The word quadratic comes from the Latin quadratus, meaning "square" — because the highest power of the variable is 2 (squared).

Quadratic equations appear constantly in science and everyday life:

The path of a thrown ball follows a parabolic (quadratic) curve
Calculating the area of a rectangle when you know the perimeter
Designing the shape of satellite dishes and car headlights
Modelling how populations grow and decline

Analogy

Imagine you throw a basketball. It goes up, reaches a peak, then comes down. The height of the ball at any moment in time follows a quadratic equation — that smooth arc is a parabola, the graph of a quadratic function.

In the equation $a x^{2} + b x + c = 0$ :

$a$ is the coefficient of $x^{2}$ (cannot be zero)
$b$ is the coefficient of $x$
$c$ is the constant term

Example

Identify $a$ , $b$ , and $c$ in the equation $3 x^{2} - 5 x + 2 = 0$ .

Solution: $a = 3$ , $b = - 5$ , $c = 2$

Remember to include the sign as part of the value!

Solutions, Roots, and the Parabola

Roots (of a quadratic)

The roots of a quadratic equation are the values of $x$ that make the equation equal to zero. They are also called solutions or zeros.

Graphically, a quadratic equation $a x^{2} + b x + c = 0$ is connected to a parabola — the U-shaped (or inverted U-shaped) curve you get when you graph $y = a x^{2} + b x + c$ .

The roots are the x-coordinates where the parabola crosses the x-axis (where $y = 0$ ).

Solutions, Roots, and the Parabola

A quadratic equation can have:

Two distinct real roots — the parabola crosses the x-axis at two points
One repeated root — the parabola just touches the x-axis at one point (the vertex)
No real roots — the parabola does not cross the x-axis at all

Note

In MYP Sciences, you will mostly encounter quadratic equations with two distinct real roots or one repeated root. Equations with no real roots still appear in some contexts (like when a thrown ball never reaches a certain height).

Exam Tip

Before solving, always rearrange your equation into standard form $a x^{2} + b x + c = 0$ first. This makes it much easier to apply any solving method.

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