Last updated: 2026-05-13

The quadratic equation

A quadratic equation is any polynomial equation of degree two: one where the highest power of the unknown is a square. The standard form is:

The coefficient $a$ determines whether the parabola opens upward ($a > 0$) or downward ($a < 0$); $b$ controls horizontal position; $c$ is where the parabola crosses the vertical axis. Setting the equation equal to zero and solving for $x$ finds the x-intercepts of that parabola — the values where the curve crosses or touches the horizontal axis.

The quadratic formula

The general solution for any quadratic equation is:

The $\pm$ symbol produces two roots, conventionally labeled $x_1$ (using $+$) and $x_2$ (using $-$). Whether those roots are real numbers, a repeated real number, or complex numbers depends entirely on the value under the square root.

The discriminant

The expression $D = b^2 - 4ac$ is called the discriminant because it discriminates between the three possible outcome types:

$D > 0$: two distinct real roots — the parabola crosses the x-axis at two separate points. Both roots can be computed directly from the quadratic formula with a real square root.

$D = 0$: one repeated real root — the parabola touches the x-axis at exactly one point (the vertex lies on the axis). The single solution is:

$D < 0$: two complex conjugate roots — the parabola does not intersect the x-axis at all. The roots involve $\sqrt{D}$ where $D$ is negative, producing an imaginary component:

Complex roots always come in conjugate pairs — $(\alpha + \beta i)$ and $(\alpha - \beta i)$ — for equations with real coefficients.

Step-by-step example

Solve $2x^2 - 5x + 2 = 0$:

1. Identify: $a = 2$, $b = -5$, $c = 2$
2. Compute discriminant: $D = (-5)^2 - 4(2)(2) = 25 - 16 = 9$
3. Since $D > 0$, two real roots exist
4. Roots: $x = \dfrac{5 \pm \sqrt{9}}{4} = \dfrac{5 \pm 3}{4}$
5. $x_1 = \dfrac{8}{4} = 2$, $\quad x_2 = \dfrac{2}{4} = 0.5$

Verify: $2(2)^2 - 5(2) + 2 = 8 - 10 + 2 = 0$ ✓

Complex root example

Solve $x^2 + x + 1 = 0$:

1. $a = 1$, $b = 1$, $c = 1$
2. $D = 1 - 4 = -3$ (negative)
3. Real part: $\alpha = -b / (2a) = -0.5$
4. Imaginary part: $\beta = \sqrt{3} / 2 \approx 0.866$
5. Roots: $x = -0.5 \pm 0.866i$

These roots have no x-axis crossing — the parabola $y = x^2 + x + 1$ has its minimum above the x-axis and never touches it.

Where quadratic equations appear

Physics — projectile motion produces quadratic equations in time: $h(t) = h_0 + v_0 t - \tfrac{1}{2}g t^2$. Setting $h = 0$ and solving for $t$ gives the time the object hits the ground.

Engineering — stress and deflection formulas, circuit analysis (second-order RLC circuits), and control systems all produce quadratic characteristic equations.

Finance — internal rate of return calculations for two-period cash flows, break-even revenue under nonlinear cost structures, and option pricing models each generate quadratics.

Geometry — finding intersection points of a line and a circle, computing distances on a coordinate plane, and solving area problems often reduce to quadratic form.

Everyday optimization — maximizing a rectangular area given a fixed perimeter, finding the dimensions that minimize material cost for a given volume, and similar problems are quadratic in structure.

Relationship to the parabola

Every quadratic equation $ax^2 + bx + c = 0$ corresponds to a parabola $y = ax^2 + bx + c$. The roots are the x-coordinates where the parabola crosses $y = 0$. Geometrically:

• Two real roots → two x-intercepts
• One real root → one x-intercept (vertex on axis)
• Complex roots → no x-intercepts (parabola entirely above or below axis)

The vertex of the parabola is at $x = -b/(2a)$, which is the average of the two real roots when they exist, and equal to the real part of the complex roots when they don't.