Why are roots (x-intercept) called the solutions of a quadratic equation?

Question

As I understand it, all the points in the hyperbole are the solutions. Although I see that it's easier to draw the curve after you discovered the roots, I don't see why to call them 'the solutions.' It has infinite solutions.

Answer 1

The equation $y=ax^2+bx+c$ has infinitely many solutions. The equation $ax^2+bx+c=0$ has (at most) two solutions, which are called the roots of the quadratic.

. . . and by the way, the first equation does not have infinite solutions. "Infinite solutions" means $x=\infty$ and/or $y=\infty$, and if we want $x$ and $y$ to be real numbers this is impossible. It is the number of solutions that is infinite, not the solutions themselves.

Analogously, "I have a large number of friends on Facebook" is not the same thing as "I have large friends on Facebook" ;-)

Answer 2

It is by definition. The x's that make the entire equation equal 0 are called the "solutions" to the equation. In other words, the x coordinates of the points where the graph intersects the x axis would be the solutions or otherwise called zeros.

Answer 3

The solutions of a quadratic $ax^2+bx+c=0$ are the values of $x$ that make the aforementioned equation true. Even if you had $ax^2+bx+c=1$, you could just subtract $1$ from both sides and that would be your new quadratic. So, they are called solutions of the quadratic because when you substitute the zeros into the quadratic equation, the equation will be satisfied.

Example: $(x+1)(x-1)=0$ has solutions $x=1$ or $x=-1$. This is because those are the only $2$ numbers that would make the equation true.

Answer 4

In addition to the excellent answers above, we typically bring $0$ on the right hand side just to factorize $ax^2+bx+c$ expression on the left hand side and then use the Zero-product property of real numbers. If you have $ax^2+bx+c=1$ and after factorization you have $a(x-r)(x-s)=1$ then you got no property to use and find the values of $x$.