What is a quadratic equation?
A quadratic equation is a second-degree equation that can be written in standard form as ax² + bx + c = 0, with a ≠ 0. Its graph is a parabola, and the solutions are the x-values where that parabola meets the x-axis.
Coefficient a
If a > 0, the parabola opens upward. If a < 0, it opens downward. A larger |a| makes the curve narrower.
Coefficient b
The coefficient b influences the horizontal position of the vertex and the axis of symmetry x = −b/(2a).
Coefficient c
The constant c is the y-intercept because f(0) = c. It tells you where the parabola crosses the vertical axis.
Roots
The roots are the x-values satisfying f(x)=0. They are the x-intercepts when they are real.
Anatomy of a quadratic graph
The quadratic formula and related equations
Δ = b² − 4ac
x = (−b ± √Δ) / 2a
h = −b / 2a
k = f(h) = −Δ / 4a
How the discriminant changes the graph
Δ > 0
There are two distinct real roots x₁ and x₂. The parabola crosses the x-axis twice.
Δ = 0
There is one repeated root. The vertex lies exactly on the x-axis.
Δ < 0
There are two complex conjugate roots and no real x-intercepts.
Worked quadratic formula examples
Write standard form
Move every term to one side so the equation becomes ax² + bx + c = 0.
Calculate Δ
Compute b² − 4ac and classify the roots from its sign.
Apply the formula
Substitute a, b, and c into (−b ± √Δ)/(2a).
Check the graph
Real roots should match the x-intercepts of the parabola.
Δ = 25 − 24 = 1
x = (5 ± 1)/2
x₁ = 3, x₂ = 2
4x² + 4x − 3 = 0
Δ = 4² − 4(4)(−3) = 64
x₁ = (−4 + 8)/8 = 0.5
x₂ = (−4 − 8)/8 = −1.5
This illustrates an important first step: before using the quadratic formula, move all terms to one side and collect them into ax² + bx + c = 0.
Δ = 16 − 16 = 0
x = −4/2
x = −2
Δ = −4
x = ±√(−4)/2
x₁ = i, x₂ = −i
Quick reference table
| Condition | Roots | Parabola |
|---|---|---|
| Δ > 0 | Two distinct real roots | Crosses x-axis twice |
| Δ = 0 | One repeated real root | Tangent to x-axis |
| Δ < 0 | Two complex conjugates | No real x-intercepts |
Quadratic formula calculator FAQ
For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac))/(2a).
Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives two complex conjugate roots.
No. If a = 0, the equation is no longer quadratic; it becomes linear (assuming b ≠ 0).
The vertex is the turning point. For y = ax² + bx + c, its x-coordinate is h = −b/(2a), and k = f(h).
When Δ < 0, the square root of the discriminant is imaginary. The roots therefore form a conjugate pair such as 2 + 3i and 2 − 3i.
Where quadratic equations appear
Projectile motion
Ignoring air resistance, height as a function of time often follows a quadratic model.
Area problems
Unknown dimensions in geometry can produce second-degree equations after expanding products.
Parabola design
Arches, reflectors, and trajectories can be modeled with quadratic functions.
Algebraic factoring
When the roots are known, the polynomial can be written as a(x − x₁)(x − x₂).