Quadratic Equations: Formula, Discriminant, and Examples

Quadratic equations are a fundamental part of high school Algebra. A mistake in calculating the discriminant can ruin the entire problem.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation where the highest degree of the variable is 2. The standard form is:

$$ax^2 + bx + c = 0$$

Where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

The Quadratic Formula and the Discriminant

To solve any quadratic equation, we use the Quadratic Formula, which relies on the Discriminant ($\Delta$).

1. Calculate the Discriminant:

$$\Delta = b^2 - 4ac$$

2. Analyze the Roots:

If $\Delta > 0$: The equation has two distinct real roots.
If $\Delta = 0$: The equation has one real root (a repeated root).
If $\Delta < 0$: The equation has no real roots (two complex roots).

3. The Quadratic Formula:

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

Step-by-Step Example

Let’s solve the equation:

$$x^2 - 5x - 6 = 0$$

Step 1: Identify coefficients

$a = 1$
$b = -5$
$c = -6$

Step 2: Calculate the Discriminant

$$\Delta = (-5)^2 - 4(1)(-6)$$

$$\Delta = 25 + 24 = 49$$

Since $\Delta > 0$, we will have two real roots.

Step 3: Apply the Quadratic Formula

$$x = \frac{-(-5) \pm \sqrt{49}}{2(1)}$$

$$x = \frac{5 \pm 7}{2}$$

Step 4: Find the solutions

$x_1 = \frac{5 - 7}{2} = \frac{-2}{2} = -1$
$x_2 = \frac{5 + 7}{2} = \frac{12}{2} = 6$

Solution: The roots are $x = -1$ and $x = 6$.

The Red Pencil: The "Double Negative" Trap

$$ b^2 - 4ac \Rightarrow -5^2 - 4(1)(-6) = -25 - 24 = -49 $$

The most common mistake isn’t forgetting the formula; it’s messing up the signs inside the square root ($\sqrt{b^2 - 4ac}$).

The $b^2$ Myth: In our example, $b = -5$. Many students write $-5^2 = -25$. Wrong. In the formula, it is $(-5)^2$, which is ALWAYS positive $25$. Any number squared (real) must be positive.
The $-4ac$ Flip: Since $c$ is negative ($-6$), the formula becomes $-4 \cdot 1 \cdot (-6)$. That double negative must turn into a plus: $+24$.

Check your math: If you get a negative number inside the square root when you expected a real solution (like $-49$), you likely missed one of these two sign flips.

If you’re struggling to identify a, b, and c correctly, review our guide on Standard Form of Quadratic Equations.

Why reading isn’t enough

Memorizing the formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$) is easy, but applying it correctly under time pressure in exams like the SAT or ACT requires practice. A simple sign error can lead to the wrong answer.

Weekzen allows you to generate infinite quadratic equations (standard, factoring, etc.) and gives you instant feedback. Whether you are in Algebra 1 or preparing for college entrance exams, the app adapts to your level to help you master the material.