Quadratic Formula
Description
The Quadratic Formula is one of the most powerful tools in algebra, providing a "master key" to solve any polynomial equation of degree 2 (a quadratic equation). Unlike factoring, which relies on trial and error and only works for simple integer roots, the quadratic formula works 100% of the time—whether the solutions are whole numbers, fractions, irrational numbers, or even imaginary numbers.
Geometrically, a quadratic equation represents a parabola. The "roots" or "solutions" calculated by this formula are simply the points where the parabola crosses the x-axis (x-intercepts). The term inside the square root, $b^2 - 4ac$, is called the **Discriminant**. It tells you the nature of the solutions: * If positive: Two distinct real solutions. * If zero: One real solution (the vertex is on the x-axis). * If negative: Two complex (imaginary) solutions.
History & Origins
The quest to solve quadratic equations dates back nearly 4000 years. Babylonians (c. 2000 BC): They could solve specific problems that we would essentially call quadratic equations today, using a "completing the square" geometric method stated in words, but they lacked a general algebraic formula. Ancient Greece (c. 300 BC): Euclid discovered a geometric approach to finding lengths that satisfy quadratic problems, though he didn't use algebra as we know it. Brahmagupta (628 AD): The Indian mathematician Brahmagupta was the first to provide an explicit, general solution in his book Brahmasphutasiddhanta. He didn't use modern notation, but his verbal recipe is equivalent to the formula we use today. Simon Stevin (1585): It wasn't until the European Renaissance that standardized notation made the formula look like the $x = ...$ version we memorize in school.
Derivation by Completing the Square
The formula comes directly from solving the general equation $ax^2 + bx + c = 0$ using the method of completing the square.
Start with the standard form: $ax^2 + bx + c = 0$
Divide everything by $a$ to get $x^2$ alone: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
Move the constant term to the right: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
Complete the square: Add $(\frac{b}{2a})^2$ to both sides. This makes the left side a perfect square.
Factor the left side: $(x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{c}{a}$
Find a common denominator on the right: $(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$
Take the square root of both sides: $x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$
Isolate $x$ by subtracting $\frac{b}{2a}$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Variables
| Symbol | Meaning |
|---|---|
a | Coefficient of x² (Quadratic term, cannot be 0) |
b | Coefficient of x (Linear term) |
c | Constant term (y-intercept) |
Examples
Basic Calculation
Problem: Solve 2x² + 5x - 3 = 0
Solution:
Physics: Projectile Motion
Problem: A ball is thrown upward. Its height h in meters after t seconds is $h(t) = -5t^2 + 20t + 2$. When does it hit the ground (h=0)?
Solution: ~4.1 seconds
- Identify coefficients from $-5t^2 + 20t + 2 = 0$: a=-5, b=20, c=2.
- Plug into formula: $t = \frac{-20 \pm \sqrt{20^2 - 4(-5)(2)}}{2(-5)}$
- Simplify discriminant: $400 - (-40) = 440$.
- Solve: $t = \frac{-20 \pm \sqrt{440}}{-10}$
- Calculate: $\sqrt{440} \approx 20.98$
- Two cases: $t = (-20 - 20.98)/-10 \approx 4.1$ and $t = (-20 + 20.98)/-10 \approx -0.1$.
- Reject negative time. The ball hits the ground at t ≈ 4.1 seconds.
Complex Roots (Imaginary Numbers)
Problem: Solve $x^2 + 4x + 5 = 0$
Solution: -2 ± i
- Identify: a=1, b=4, c=5.
- Discriminant: $b^2 - 4ac = 16 - 4(1)(5) = 16 - 20 = -4$.
- Since the discriminant is negative, there are no real roots.
- Square root of -4 is $2i$.
- Formula: $x = \frac{-4 \pm 2i}{2}$
- Simplify: $x = -2 \pm i$.
Common Mistakes
Wrong sign for -b
If b is negative (e.g., -5), then -b becomes positive (+5). A common error is writing the wrong sign.
Dividing only the first part by 2a
The fraction bar goes under the ENTIRE numerator, not just the square root part. Correct: $\frac{A+B}{C}$. Incorrect: $A + \frac{B}{C}$.
Forgetting the ±
Quadratic equations usually have two answers. Forgetting the "plus or minus" means you lose half your solution.
Real-World Applications
Ballistics & Sports
Any object thrown or fired under gravity follows a parabolic arc. The quadratic formula calculates exactly how long the object stays in the air and where it will land.
Economics & Profit Maximization
Profit usually follows a curve—profit increases with price up to a point, then decreases as fewer people buy. This curve is a quadratic equation. Finding the vertex or roots helps businesses set optimal prices or break-even points.
Architecture & Engineering
Usage of parabolic shapes in suspension bridges and antenna dishes. Engineers use quadratic equations to calculate stress loads and focal points.
Frequently Asked Questions
Why is it called "Quadratic" if quad means 4?
It comes from the Latin word "quadratus" (square), because the highest power is squared ($x^2$). It refers to the 4 sides of a square, not the number 4 itself.
When should I use factoring instead?
If the numbers are small and simple integers (like $x^2 + 5x + 6$), factoring is faster. Use the formula for messy numbers, decimals, or when you get stuck.