why do we call quadratics equation by that name?

Question

I'm still learning about this so any clarification would be very helpful

1. What is the purpose of quadratics equations?
2. How did Mathematician come up with it?
3. How did they figure out the formula for solving quadratics equations?

Answer 1

In order:

1) Quadratic equations have all sorts of applications. The most natural is the arc of a thrown ball: you can write a quadratic equation $y = at^2 + bt + c$ so that $y$ is the height of the ball above the ground at time $t$. But there are millions of other applications - seriously, just Google "applications of quadratic equations" and you'll get millions of results.

2) Mathematicians came up with them because they're very natural. Multiplication and addition come naturally to us, so the easiest kind of equation we thought of was linear equations - like $4x + 3$. The second-easiest were the ones that required just one multiplication of variables, which means a quadratic.

3) The proof is based on a technique called completing the square. It works like this:

$ax^2 + bx + c = 0$

$x^2 + (b/a)x + c/a = 0$ (divide both sides by $a$)

$(x + \frac{b}{2a})^2 + c/a - \frac{b^2}{4a^2} = 0$ (a clever trick; try expanding out the left side of the equation and you'll see it's the same as the previous line)

$(x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{c}{a}$ (move stuff to the right-hand side)

$x + \frac{b}{2a} = \pm\sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}$ (undo the square on both sides)

$x = \frac{-b}{2a} \pm \sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}$ (move the last piece over)

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (simplify the right-hand side to make it look like the usual formula)