Mathematics for Physics I - Quadratic Equations

Standard Form and Solutions

Quadratic equations are second-degree polynomial equations that appear frequently in physics problems involving motion, energy, and oscillations.

Standard Form

\[ ax^2 + bx + c = 0 \]

Where:

$a \neq 0$ (otherwise the equation would be linear)
$a$, $b$, and $c$ are constants, typically real numbers
$x$ is the variable we're solving for

Quadratic Formula

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

The quadratic formula gives the two solutions (roots) of the quadratic equation.

The expression under the square root, $b^2 - 4ac$, is called the discriminant and determines the nature of the solutions:

If $b^2 - 4ac > 0$, there are two distinct real solutions
If $b^2 - 4ac = 0$, there is one repeated real solution
If $b^2 - 4ac < 0$, there are two complex conjugate solutions

Interactive visualization of quadratic functions and their roots

Alternative Solution Methods

Completing the Square

This method transforms the quadratic equation into a perfect square plus a constant:

\begin{align} ax^2 + bx + c &= 0 \\ x^2 + \frac{b}{a}x + \frac{c}{a} &= 0 \\ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} &= 0 \\ \left(x + \frac{b}{2a}\right)^2 &= \frac{b^2}{4a^2} - \frac{c}{a} \\ \left(x + \frac{b}{2a}\right)^2 &= \frac{b^2 - 4ac}{4a^2} \\ x + \frac{b}{2a} &= \pm\frac{\sqrt{b^2 - 4ac}}{2a} \\ x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{align}

This approach is particularly useful for deriving the quadratic formula and for understanding the geometric interpretation of quadratic equations.

Factoring

If the quadratic expression can be factored, we can write:

\[ ax^2 + bx + c = a(x - r_1)(x - r_2) = 0 \]

Where $r_1$ and $r_2$ are the roots of the equation.

This leads to the solutions $x = r_1$ or $x = r_2$.

The relationship between the coefficients and roots is:

\[ r_1 + r_2 = -\frac{b}{a} \] \[ r_1 \cdot r_2 = \frac{c}{a} \]

These are known as Vieta's formulas for quadratic equations.

Graphical Interpretation

The graph of a quadratic function $f(x) = ax^2 + bx + c$ is a parabola:

Key Features

Vertex: The highest or lowest point of the parabola, located at $x = -b/(2a)$
Axis of Symmetry: The vertical line passing through the vertex
Direction: Opens upward if $a > 0$, downward if $a < 0$
y-intercept: The point $(0, c)$
x-intercepts: The points where the parabola crosses the x-axis, corresponding to the roots of the quadratic equation

Vertex Form

The quadratic function can be written in vertex form:

\[ f(x) = a(x - h)^2 + k \]

Where $(h, k)$ is the vertex of the parabola.

The relationship with standard form is:

\[ h = -\frac{b}{2a} \] \[ k = c - \frac{b^2}{4a} \]

This form is particularly useful for identifying the maximum or minimum value of the quadratic function.

Interactive visualization of parabolas and their key features

Applications in Physics

Projectile Motion

The position of a projectile under constant gravity follows a quadratic equation:

\[ y = y_0 + v_0\sin(\theta)t - \frac{1}{2}gt^2 \]

The horizontal range R of a projectile launched from ground level is:

\[ R = \frac{v_0^2\sin(2\theta)}{g} \]

This is maximized when θ = 45°.

Simple Harmonic Motion

The energy of a simple harmonic oscillator is described by a quadratic equation:

\[ E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 \]

The period of oscillation is:

\[ T = 2\pi\sqrt{\frac{m}{k}} \]

This is independent of the amplitude, a key feature of simple harmonic motion.

Optics

The shape of parabolic mirrors is described by quadratic equations. These mirrors have the property that all rays parallel to the axis are reflected to the focus:

\[ y = \frac{x^2}{4f} \]

Where f is the focal length of the mirror.

This property is used in telescopes, satellite dishes, and solar concentrators.

Quantum Mechanics

The potential energy of a quantum harmonic oscillator is quadratic:

\[ V(x) = \frac{1}{2}kx^2 \]

The energy levels are quantized as:

\[ E_n = \hbar\omega\left(n + \frac{1}{2}\right) \]

Where ω = √(k/m) and n is a non-negative integer.

Example: Projectile Motion Analysis

Consider a ball thrown with initial velocity v₀ = 20 m/s at an angle θ = 30° from the horizontal. The position of the ball as a function of time is given by:

\[ x(t) = (v_0\cos\theta)t = (20\cos30°)t = 17.32t \] \[ y(t) = (v_0\sin\theta)t - \frac{1}{2}gt^2 = (20\sin30°)t - 4.9t^2 = 10t - 4.9t^2 \]

To find when the ball hits the ground, we solve y(t) = 0:

\[ 10t - 4.9t^2 = 0 \] \[ t(10 - 4.9t) = 0 \]

This gives t = 0 (initial position) or t = 10/4.9 ≈ 2.04 seconds (when the ball lands).

The horizontal range is therefore:

\[ R = x(2.04) = 17.32 \times 2.04 \approx 35.3 \text{ meters} \]

Quadratic Optimization Problems

Many optimization problems in physics involve finding the maximum or minimum value of a quadratic function.

Finding Extrema

For a quadratic function $f(x) = ax^2 + bx + c$:

The critical point occurs at $x = -b/(2a)$
If $a > 0$, this is a minimum
If $a < 0$, this is a maximum
The extreme value is $f(-b/(2a)) = c - b^2/(4a)$

Example: Minimum Energy

A spring-mass system has potential energy $U(x) = \frac{1}{2}kx^2$ and kinetic energy $K(v) = \frac{1}{2}mv^2$.

The total energy is $E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$.

For a given energy E, the maximum displacement occurs when v = 0, giving:

\[ x_{max} = \pm\sqrt{\frac{2E}{k}} \]

This represents the amplitude of oscillation.

Key Insight:

Quadratic equations are fundamental in physics because many natural phenomena involve forces or energies that are proportional to the square of a variable. The parabolic shape of quadratic functions describes the trajectory of projectiles, the energy of oscillators, and many other physical systems. Understanding how to analyze and solve quadratic equations is essential for modeling and predicting the behavior of these systems.

Practice Problems

Test your understanding of quadratic equations with these practice problems:

A ball is thrown upward from a height of 1.5 meters with an initial velocity of 12 m/s. Find when the ball will hit the ground. (Use g = 9.8 m/s²)
Find the values of p for which the quadratic equation x² + px + 1 = 0 has equal roots.
A rectangular garden with area 200 m² is to be enclosed by a fence. If the fencing material for the front costs $10 per meter and for the other three sides costs $5 per meter, find the dimensions that minimize the cost.
A projectile is launched at an angle θ from the horizontal with initial speed v₀. Show that the maximum height reached is h = (v₀²sin²θ)/(2g).
For the quadratic function f(x) = 2x² - 8x + 9, find the vertex, axis of symmetry, y-intercept, and x-intercepts (if any).