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 word for "square" — which makes sense, because the highest power of the variable is 2 (squared).

Quadratic equations appear everywhere in the real world:

The path of a thrown ball follows a quadratic curve
Architects use quadratics when designing arches
Physicists use them to calculate falling distances

Note

Every quadratic equation can have up to two solutions (also called roots). Sometimes both solutions are real numbers, sometimes they are equal, and sometimes there are no real solutions at all.

Before we learn the methods to solve them, let's get comfortable recognising quadratic equations:

$x^{2} - 5 x + 6 = 0$ ✓ (standard form)
$2 x^{2} + 3 x = 0$ ✓ (c = 0)
$x^{2} = 9$ ✓ (can be rearranged)
$x^{3} + 2 x = 0$ ✗ (this is cubic, degree 3)

Simple Quadratics: Equations of the Form x² = k

The simplest type of quadratic equation has no $x$ term — just $x^{2}$ on one side and a number on the other.

Consider $x^{2} = 7$ . We are asking: "What number, when multiplied by itself, gives 7?"

There are actually two answers:

$7 \times 7 = 7$ ✓
$(- 7) \times (- 7) = 7$ ✓

So we write: $x = \pm 7$

The symbol $\pm$ means "plus or minus" — it's a compact way of writing two solutions at once.

± (Plus or Minus)

The symbol $\pm$ indicates two values: one positive and one negative. For example, $x = \pm 5$ means $x = 5$ or $x = - 5$ .

Example

Solve $x^{2} = 25$

Step 1: Take the square root of both sides.
$x = \pm 25$
Step 2: Simplify.
$x = \pm 5$

So $x = 5$ or $x = - 5$ . ✓

Check: $5^{2} = 25$ ✓ and $(- 5)^{2} = 25$ ✓

Warning

A very common mistake is to write only the positive answer. Always remember: when you square root both sides, you get two solutions — one positive and one negative!

Note

What if $k$ is negative? For example, $x^{2} = - 4$ has no real solutions, because no real number multiplied by itself can give a negative result. You will see this idea again when we study the discriminant later in these notes.

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