A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where x is the unknown you’re solving for and a, b, and c are known numbers. The one rule: a cannot equal zero. If it were, the x² term would disappear and you’d just have a linear equation (a straight line). The word “quadratic” comes from the Latin quadratus, meaning “square,” because the variable is squared.
In this standard form, a is called the quadratic coefficient, b is the linear coefficient, and c is the constant term. These three numbers completely define the equation and determine everything about its solutions and its graph.
How to Solve With the Quadratic Formula
The most universal way to solve any quadratic equation is the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
You plug in your values for a, b, and c, and the formula hands you the solutions. The “±” symbol means you calculate it twice: once with a plus sign and once with a minus sign. That gives you up to two solutions, which makes sense because a parabola can cross the horizontal axis in two places, one place, or not at all.
What the Discriminant Tells You
The expression under the square root sign, b² – 4ac, is called the discriminant. It’s a quick way to know what kind of solutions you’ll get before you finish the calculation:
- Positive discriminant: two distinct real solutions. The parabola crosses the x-axis at two points.
- Zero discriminant: one repeated real solution. The parabola just touches the x-axis at its vertex.
- Negative discriminant: no real solutions. Instead, you get two complex solutions involving the imaginary unit i (defined as the square root of -1). The parabola never touches the x-axis.
For example, with the equation x² + 49 = 0, the discriminant is negative. Solving it gives x = ±7i. These aren’t numbers on the regular number line, but they matter in engineering and physics. A complex number takes the form a + bi, where a is the real part and b is the imaginary part.
Other Ways to Solve Quadratics
Factoring
If the equation breaks apart neatly into two binomials, factoring is the fastest method. For instance, x² + 5x + 6 = 0 factors into (x + 2)(x + 3) = 0, giving solutions x = -2 and x = -3. The catch is that many quadratic equations don’t factor into clean whole numbers, so this method only works in certain cases.
Completing the Square
This method rewrites the equation so one side becomes a perfect square. Take x² + 6x = -2 as an example. You add 9 to both sides (because (6/2)² = 9), turning the left side into (x + 3)². Now you have (x + 3)² = 7, and taking the square root of both sides gives x = -3 ± √7. The general trick: for any expression x² + bx, you add (b/2)² to complete the square. If the coefficient on x² isn’t 1, divide everything by that coefficient first.
Completing the square is actually how the quadratic formula itself is derived. It works every time, though the arithmetic can get messy.
The Parabola: What Quadratics Look Like
When you graph a quadratic equation as y = ax² + bx + c, the result is always a parabola, a U-shaped curve. If a is positive, the parabola opens upward. If a is negative, it opens downward.
The most important point on a parabola is the vertex, its highest or lowest point. You find the x-coordinate of the vertex with x = -b / (2a), then plug that value back into the equation to get the y-coordinate. The vertical line passing through the vertex, x = -b/(2a), is the axis of symmetry. The parabola is a mirror image on either side of this line. The constant c tells you where the parabola crosses the y-axis.
Why Quadratics Show Up Everywhere
Quadratic equations model any situation where something changes at a rate that is itself changing, which turns out to be surprisingly common.
Projectile motion is the classic example. When you throw a ball, kick a soccer ball, or launch fireworks, the height of the object over time follows a quadratic equation. The path through the air traces a parabola. In physics, the height equation typically looks like y = (initial upward velocity)t – ½gt², where g is the acceleration due to gravity and t is time. Solving this as a quadratic lets you calculate when the object hits the ground, how high it goes, or how long it stays in the air. One physics textbook example calculates a tennis ball’s flight by rearranging the trajectory into 4.9t² – 21.2t + 10 = 0 and applying the quadratic formula to find the exact time it reaches a spectator.
Optimization problems also rely on quadratics. Because a parabola has a single peak (if it opens downward) or a single valley (if it opens upward), the vertex gives you the maximum or minimum value of whatever you’re modeling. If you’re fencing a rectangular garden with a fixed amount of fencing and want the largest possible area, the area equation is quadratic. Finding the vertex tells you the dimensions that maximize the space. In one example, an area function of A = 80x – 2x² reaches its maximum at x = 20, yielding a maximum area of 800 square units.
These equations also appear in economics (profit and cost modeling), architecture (the shape of arches and cables), and anywhere parabolic curves naturally occur.
A Brief History
People were solving quadratic equations long before algebra existed as a subject. Babylonian mathematicians around 2000 BCE worked out solutions on clay tablets using a method that resembles completing the square, driven by practical needs like measuring land and planning construction. Indian scholars described geometric solutions in the Sulba Sutras around 600 BCE, developed for the purpose of building altars. Euclid formalized these ideas using geometric constructions around 300 BCE, and Diophantus around 200 CE moved toward something more recognizable as algebra by introducing symbols for unknowns and working with numerical solutions. The symbolic formula we use today is the endpoint of roughly 4,000 years of mathematical development.