Quadratic Equations — Quadratic Formula, Discriminant, and Examples

A quadratic equation is an equation that can be written as $ax^2 + bx + c = 0$ under the condition $a \ne 0$:

$ax^2 + bx + c = 0 \quad (a \ne 0)$

Here $a$, $b$, and $c$ are the coefficients, and the highest power of the variable must be $2$. For example, $x^2 - 5x + 6 = 0$ is a quadratic equation, but $2x + 3 = 0$ is a linear equation. Solving a quadratic equation means finding the value(s) of $x$ that make the equation true, which visually corresponds to finding where the parabola $y = ax^2 + bx + c$ intersects the $x$-axis.

Why standard form comes first

Before you start solving, align the equation into standard form:

$ax^2 + bx + c = 0$

This form makes it harder to misread the signs of $a$, $b$, and $c$, and it is especially important not to skip when using the quadratic formula. Once the equation is organized, the discriminant gives a quick preview of the answer. The discriminant is:

$D = b^2 - 4ac$

Because $D = b^2 - 4ac$ sits inside the square root of the quadratic formula, its sign directly determines the number of real solutions:

• If $D > 0$, there are two distinct real solutions.
• If $D = 0$, there is one real solution (a repeated root).
• If $D < 0$, there are no real solutions.

This explanation assumes real numbers. If you extend your scope to complex numbers, solutions exist even when $D < 0$.

The calculation procedure

Once in standard form, choose your approach:

1. If you can factor it immediately, use factorization first, since it is fastest.
2. If you want to make the structure clearer, consider completing the square.
3. If factorization isn't obvious but you want a guaranteed result, use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The basic flow is the same every time: check the number of solutions with the discriminant, then find the actual values with the quadratic formula.

Worked example: discriminant and formula together

Solve

$x^2 - 4x - 1 = 0$

Since it's already in standard form, $a = 1$, $b = -4$, and $c = -1$. Factorization isn't immediately obvious, so we use the quadratic formula.

First calculate the discriminant:

$D = b^2 - 4ac = (-4)^2 - 4(1)(-1) = 16 + 4 = 20$

Since $D > 0$, there are two real solutions. Substitute into the formula:

$x = \frac{-(-4) \pm \sqrt{20}}{2 \cdot 1} = \frac{4 \pm \sqrt{20}}{2}$

Since $\sqrt{20} = 2\sqrt{5}$:

$x = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5}$

Therefore the solutions are:

$x = 2 + \sqrt{5},\quad x = 2 - \sqrt{5}$

Check with the original equation:

$\left(2 + \sqrt{5}\right)^2 - 4\left(2 + \sqrt{5}\right) - 1 = 0$

The same result $0$ is obtained for the other solution.

Practice to solidify the steps

Try solving this on your own:

$x^2 + 2x - 3 = 0$

Start by checking the standard form, then see if it factors. If that's too difficult, switch to the quadratic formula. Following this order keeps your process organized. For one more, choose an equation that is hard to factor and use the discriminant and quadratic formula as a set, so the choice of method becomes natural.

Calculation traps to watch for

Reading coefficients without standardizing

If you leave the equation as $x^2 = 4x + 1$, it's easy to mix up the signs of $b$ and $c$. Convert it to

$x^2 - 4x - 1 = 0$

before proceeding.

Calculating only one of the $\pm$

In the quadratic formula you must calculate both $+$ and $-$. If you've written only one answer for a quadratic equation, check this part first.

Misunderstanding the discriminant

The discriminant is not the solution itself, but a value used to judge the number of real solutions. After calculating $D$, you then proceed to the quadratic formula or factorization.

Where quadratic equations appear

In school mathematics, quadratic equations frequently appear in problems involving parabolas, maximum and minimum values, area conditions, and formulas for speed or motion. They are fundamental tools whenever you solve relationships involving $x^2$. The method you choose depends on the form of the equation: factorization is fastest if possible; otherwise, the quadratic formula is the most reliable.