What is a quadratic equation? (We explain on balls)
Imagine that you threw a basketball into a hoop. It flies not in a straight line, but in a beautiful arc, rising up and falling down. In mathematics this arc is called aparabola. And to calculate exactly where the ball will land, mathematicians use quadratic equations.
A Quadratic equation is an equation in which the most important (unknown) number is multiplied by itself, that is, raised to the second power (squared). Hence the name!
ax2 + bx + c = 0
Parsing the equation piece by piece (Coefficients)
In our main formula, the letters a, b and c are just ordinary numbers called coefficients. And the letter x is the very unknown (the root of equation) that we need to find as detectives.
| Letter | What is it called? | Main rule |
|---|---|---|
| a | High coefficient | Never equal to zero! (Otherwise x2 will disappear) |
| b | Second coefficient | Can be any number, even zero. Stands next to the regular x. |
| c | Free member | A lonely number without any X's. Can be anything. |
Example from life: In the equation 2x2 − 5x + 3 = 0 the numbers are distributed like this: a = 2, b = -5, c = 3.
Main tool: What is the Discriminant?
In the old days, mathematicians racked their brains for hours to find the answer. But then they came up with a universal tool that looked like a magic wand. It is called Discriminant (denoted by a capital letter D). This is a special hint number that tells us in advance how many answers we will get.
The discriminant formula is very simple (it is taught by heart in all schools in the world):
D = b2 − 4ac
🔥 Three golden rules of Discriminant (Required know!)
Once you have calculated D, look at the result:
- If D is greater than zero (D > 0): Hurray! The equation has two different answers (two roots). A parabola intersects the earth in two places.
- If D is zero (D = 0):The equation has only one answer. The ball just touched the ground and bounced.
- If D is less than zero (D < 0): Stop the game! In school mathematics they say that there are no roots (it is impossible to solve). The graph hangs in the air and does not touch the ground.
How to find the answers (roots) yourself?
If the discriminant allows us to decide further (it is greater than or equal to zero), we use the final formula to find our x's (roots):
The ± sign means that we count the formula twice: the first time we add the root from the discriminant, and the second time we subtract it. This is how we get our two treasured answers!
Why is this necessary in the modern world?
It seems like these are just boring school formulas. But in fact, quadratic equations are the engine of modern technology!
- 🚀 Physics and space: Calculating the flight path of a SpaceX rocket, a thrown stone or a bullet in a video game is impossible without these formulas.
- 🏗️ Architecture: Beautiful suspension bridges and arches are built strictly according to the laws of the parabola.
- 💻 Calculator programming: These are the algorithms hidden “under the hood” of smart Client-Side converters and online calculators on the pages of our website. The computer instantly calculates the discriminant in a fraction of a second and gives you the finished result!