Zeros of a Quadratic Formula

Zeros of a quadratic is the zeros (or roots) of a quadratic function f(x) = ax^2 + bx + c are the values of x where f(x) = 0.

The Formula

For ax2+bx+c=0ax^2 + bx + c = 0: zeros are x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Sum of zeros =βˆ’ba= -\frac{b}{a}, product of zeros =ca= \frac{c}{a}.

When to use: The zeros are where the parabola crosses or touches the xx-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)=x2βˆ’5x+6=(xβˆ’2)(xβˆ’3)f(x) = x^2 - 5x + 6 = (x - 2)(x - 3)
Zeros: x=2x = 2 and x=3x = 3. The parabola crosses the xx-axis at (2,0)(2, 0) and (3,0)(3, 0).

Notation

Zeros are also called roots or xx-intercepts. Written as r1r_1, r2r_2 or x1x_1, x2x_2. Graphically, they are the points (r,0)(r, 0) where the parabola meets the xx-axis.

What This Formula Means

The zeros (or roots) of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c are the values of xx where f(x)=0f(x) = 0. Graphically, they are the xx-intercepts of the parabola.

The zeros are where the parabola crosses or touches the xx-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)=ax2+bx+cf(x) = ax^2 + bx + c are Z(f)={x∈R∣f(x)=0}Z(f) = \{x \in \mathbb{R} \mid f(x) = 0\}. By the quadratic formula, Z(f)={βˆ’bΒ±b2βˆ’4ac2a}∩RZ(f) = \left\{\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\right\} \cap \mathbb{R}, with ∣Z(f)∣∈{0,1,2}|Z(f)| \in \{0, 1, 2\}.

Worked Examples

Example 1

easy
Find the zeros of f(x)=x2βˆ’7x+10f(x) = x^2 - 7x + 10.

Answer

x=2Β andΒ x=5x = 2 \text{ and } x = 5

First step

1
Set f(x)=0f(x) = 0: x2βˆ’7x+10=0x^2 - 7x + 10 = 0.

Full solution

  1. 2
    Factor: (xβˆ’2)(xβˆ’5)=0(x - 2)(x - 5) = 0.
  2. 3
    Zeros: x=2x = 2 and x=5x = 5.
The zeros of a function are the xx-values where f(x)=0f(x) = 0. They correspond to the xx-intercepts of the graph.

Example 2

medium
Find the zeros of g(x)=2x2βˆ’8xg(x) = 2x^2 - 8x.

Example 3

medium
Find the zeros of f(x)=x2+6x+5f(x) = x^2 + 6x + 5 by factoring.

Common Mistakes

  • Taking only the ++ branch of Β±\pm - a quadratic generally has two zeros; keep both signs.
  • Reporting the zeros with flipped signs from factors - (xβˆ’3)(x-3) gives zero 33, (x+5)(x+5) gives zero βˆ’5-5.
  • Calling 'no real zeros' an error - when Ξ”<0\Delta<0 the parabola simply does not cross the x-axis.

Why This Formula Matters

Zeros are the solutions to the quadratic equation itself, the answers to projectile-lands, break-even, and intersection problems. They also reconstruct the equation through sum =βˆ’ba=-\tfrac{b}{a} and product =ca=\tfrac{c}{a}. Recognizing it by "Am I looking for the x-values that make the quadratic equal zero?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from y-intercept and vertex and discriminant in a mixed problem set.

Frequently Asked Questions

What is the Zeros of a Quadratic formula?

The zeros (or roots) of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c are the values of xx where f(x)=0f(x) = 0. Graphically, they are the xx-intercepts of the parabola.

How do you use the Zeros of a Quadratic formula?

The zeros are where the parabola crosses or touches the xx-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 xx-intercepts. Written as r1r_1, r2r_2 or x1x_1, x2x_2. Graphically, they are the points (r,0)(r, 0) where the parabola meets the xx-axis.

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

Zeros are the solutions to the quadratic equation itself, the answers to projectile-lands, break-even, and intersection problems. They also reconstruct the equation through sum =βˆ’ba=-\tfrac{b}{a} and product =ca=\tfrac{c}{a}. Recognizing it by "Am I looking for the x-values that make the quadratic equal zero?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from y-intercept and vertex and discriminant in a mixed problem set.

What do students get wrong about Zeros of a Quadratic?

The procedure for zeros of a quadratic is the easy part; the trap is taking only the ++ branch of Β±\pm. Asking "Am I looking for the x-values that make the quadratic equal zero?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 β†’
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