Graph and Roots of Quadratic Polynomial

A quadratic equation

ax² + bx + c = 0,

with the leading coefficient a ≠ 0, has two roots that may be real - equal or different - or complex. The roots can be found from the quadratic formula:

x_1,2 = (-b ± √b² - 4ac) / 2a,

In addition to the four arithmetic operations, the formula includes a square root. The expression under the square root, D = b² - 4ac - known as the discriminant - may be positive, zero, or negative. Correspondingly, the equation may have two real and distinct roots, two equal real roots, or two conjugate complex roots.

The applet allows one to experiment by changing the coefficients of the polynomial by dragging the three scrollbars at the top of the applet.)

(The maximum and minimum values on the axes can also be changed by clicking at the numbers at the endpoints - a little off their vertical mid-axis.)

1 November 2015, Created with GeoGebra

The graph of the quadratic polynomial is a parabola, with the horns pointing upwards if a > 0 or downwards if a < 0.

Observe on the graph the behavior of the roots as you change the other two coefficients, b and c. If real, they lie on the x-axis symmetrically with respect to the vertical line through the parabola's extreme point. If complex, they simply lie on that line symmetrically with respect to the x-axis.

If x₁ and x₂ are the roots of a quadratic polynomial P(x), the polynomial is factored as in

P(x) = a(x - x₁)(x - x₂)

because both sides of the identity are quadratic polynomials that vanish at x₁ and x₂ and have the same leading coefficient. Multiplying through we get

ax² + bx + c = ax² - a(x₁ + x₂)x + ax₁x₂

implying the equality of coefficients:

b = - a(x₁ + x₂) and
c = ax₁x₂.

We thus obtain a theorem by François Viète (1540 - 1603) for quadratic equations:

x₁ + x₂ = - b / a and
x₁x₂ = c / a

which are known as Viète's formulas. These tells us something about how the roots of the polynomial change when a and c remain constant.

The interesting case is when ac > 0. This is when the discriminant may become negative leading to complex roots. If at the outset the roots are real then, as b changes, the roots move in opposite directions alongside the x-axis. If the movement is towards each other, they coalesce momentarily and then turn complex until they coalesce again and become real once more. What is interesting is their trajectory while complex. Since complex roots of a quadratic polynomial with real coefficients are conjugate, say x = α ± i β, their product equals the square of the modulus:

(α + i β)(α - i β) = α² + β²

so that the module of the roots remain constant and equal to √c/a. This means that complex roots traverse a circle of that radius centered at the origin!

When c changes, the behavior of the graph is less exciting: it just goes up for a positive change in c and down for a negative change. But, if you start with an upward graph with two real roots and keep increasing c then, at some point, viz., for the value of c for which ax² + bx + c is a complete square, the graph will become tangent to the x-axis meaning that the equation has two equal (and necessarily real) roots. After that, the roots split into a conjugate pair.

Quadratic Formula and Equation

Geometric Construction of Roots of Quadratic Equation
Quadratic Formula - Completing And Not Completing the Square
Graph and Roots of Quadratic Polynomial
Emergence of Chaos
Maximum Volume of a Cut Off Box
Golden Ratio in Geometry
Viète's Formulas

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