Zeros of a Quadratic Formula

The Formula

For ax^2 + bx + c = 0: zeros are x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Sum of zeros = -\frac{b}{a}, product of zeros = \frac{c}{a}.

When to use: The zeros are where the parabola crosses or touches the x-axis. A parabola can cross twice (two zeros), just touch once (one repeated zero), or miss entirely (no real zeros). You can find them by factoring, completing the square, or using the quadratic formula.

Quick Example

f(x) = x^2 - 5x + 6 = (x - 2)(x - 3)
Zeros: x = 2 and x = 3. The parabola crosses the x-axis at (2, 0) and (3, 0).

Notation

Zeros are also called roots or x-intercepts. Written as r_1, r_2 or x_1, x_2. Graphically, they are the points (r, 0) where the parabola meets the x-axis.

What This Formula Means

The zeros (or roots) of a quadratic function f(x) = ax^2 + bx + c are the values of x where f(x) = 0. Graphically, they are the x-intercepts of the parabola.

The zeros are where the parabola crosses or touches the x-axis. A parabola can cross twice (two zeros), just touch once (one repeated zero), or miss entirely (no real zeros). You can find them by factoring, completing the square, or using the quadratic formula.

Formal View

The zeros of f(x) = ax^2 + bx + c are Z(f) = \{x \in \mathbb{R} \mid f(x) = 0\}. By the quadratic formula, Z(f) = \left\{\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\right\} \cap \mathbb{R}, with |Z(f)| \in \{0, 1, 2\}.

Worked Examples

Example 1

easy
Find the zeros of f(x) = x^2 - 7x + 10.

Solution

  1. 1
    Set f(x) = 0: x^2 - 7x + 10 = 0.
  2. 2
    Factor: (x - 2)(x - 5) = 0.
  3. 3
    Zeros: x = 2 and x = 5.

Answer

x = 2 \text{ and } x = 5
The zeros of a function are the x-values where f(x) = 0. They correspond to the x-intercepts of the graph.

Example 2

medium
Find the zeros of g(x) = 2x^2 - 8x.

Common Mistakes

  • Forgetting that a quadratic can have 0, 1, or 2 real zeros depending on the discriminant
  • Confusing zeros (x-values where f(x) = 0) with the y-intercept (f(0), which is the value at x = 0)
  • Only finding one zero when there are two—always check for the \pm from the quadratic formula

Why This Formula Matters

Finding zeros is one of the most fundamental tasks in algebra. Zeros appear in physics (when does the ball hit the ground?), economics (break-even points), and throughout higher mathematics.

Frequently Asked Questions

What is the Zeros of a Quadratic formula?

The zeros (or roots) of a quadratic function f(x) = ax^2 + bx + c are the values of x where f(x) = 0. Graphically, they are the x-intercepts of the parabola.

How do you use the Zeros of a Quadratic formula?

The zeros are where the parabola crosses or touches the x-axis. A parabola can cross twice (two zeros), just touch once (one repeated zero), or miss entirely (no real zeros). You can find them by factoring, completing the square, or using the quadratic formula.

What do the symbols mean in the Zeros of a Quadratic formula?

Zeros are also called roots or x-intercepts. Written as r_1, r_2 or x_1, x_2. Graphically, they are the points (r, 0) where the parabola meets the x-axis.

Why is the Zeros of a Quadratic formula important in Math?

Finding zeros is one of the most fundamental tasks in algebra. Zeros appear in physics (when does the ball hit the ground?), economics (break-even points), and throughout higher mathematics.

What do students get wrong about Zeros of a Quadratic?

Choosing the right method: try factoring first; if that fails, use the quadratic formula. The discriminant tells you in advance how many real zeros to expect.

What should I learn before the Zeros of a Quadratic formula?

Before studying the Zeros of a Quadratic formula, you should understand: quadratic functions, factoring, quadratic formula.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
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