Quadratics | FE Mechanical Analytic geometry

Quadratics

Quadratic equations represent one of the most important classes of polynomial equations in mathematics and engineering. These equations appear frequently in physics, engineering, and other applied sciences.

The standard form of a quadratic equation is:

$a x^{2} + b x + c = 0,$

where $a$ , $b$ , and $c$ are constants with $a \neq = 0$ .

Roots of quadratic equations

The solutions (roots) of a quadratic equation are given by the quadratic formula:

Quadratic formula

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

This formula provides the exact solutions for any quadratic equation and is derived through the process of completing the square. The expression under the square root, called the discriminant, is a powerful tool for analyzing quadratic equations without actually solving them.

Definitions

Discriminant: The expression $D = b^{2} - 4 a c$ under the square root in the quadratic formula. Its sign determines the nature of a quadratic equation’s roots: two distinct real roots when $D > 0$ , one repeated real root when $D = 0$ , and two complex conjugate roots when $D < 0$ .

Case 1: $D > 0$ , two distinct real roots. The quadratic equation has two different real solutions. Graphically, the parabola crosses the $x$ -axis at two distinct points.
Case 2: $D = 0$ , one repeated real root. There is exactly one real solution. The parabola touches the $x$ -axis at exactly one point (the vertex).
Case 3: $D < 0$ , two complex conjugate roots. The equation has no real solutions, but two complex solutions that are conjugates of each other. The parabola does not intersect the $x$ -axis.

Completing the square

Completing the square is a technique for solving quadratic equations and understanding their geometric properties. This method transforms a quadratic expression into perfect square form.

Starting with $a x^{2} + b x + c = 0$ where $a \neq = 0$ :

Factor out the leading coefficient: $a (x^{2} + \frac{b}{a} x) + c = 0$
Complete the square inside parentheses: add and subtract $(\frac{b}{2 a})^{2}$
Rewrite as a perfect square: $a (x + \frac{b}{2 a})^{2} - \frac{b ^{2}}{4 a} + c = 0$
Solve for $x$ : $a (x + \frac{b}{2 a})^{2} = \frac{b ^{2} - 4 a c}{4 a}$

This process directly leads to the quadratic formula and reveals important geometric properties, as shown in the figure below:

@@@ Figure [quadratics-1]: A quadratic function with vertex $(h, k)$ and axis of symmetry $x = - \frac{b}{2 a}$ @@@

Vertex form of quadratics

Completing the square transforms a quadratic into vertex form:

$y = a (x - h)^{2} + k,$

where $(h, k)$ is the vertex of the parabola. This form immediately reveals:

Vertex location: $(- \frac{b}{2 a}, f (- \frac{b}{2 a}))$
Axis of symmetry: $x = - \frac{b}{2 a}$
Direction of opening: upward if $a > 0$ , downward if $a < 0$