Can a quadratic equation not equal to zero

Question

Well I wanted to know whether or not $y = x^2 + x + 7$ is a quadratic equation since the general form is $ax^2 + bx + c = 0$ here the equation $y=x^2+x+7$. Isn't equal to zero so I'm a bit confused

Answer 1

The general form is not $ax^2+bx+c=0$, but $y = ax^2+bx+c$. To have a function, you must be able to put one number in ($x$ in this case), and output one number out ($y$).

$ax^2+bx+c =0$ when $y = 0$, so when you write this, you are trying to find which $x$ values the function outputs $0$ (or the roots). This doesn't help you figure out if $y=x^2+x+7$ is a quadratic equation.

Answer 2

When you write $y=x^2 + x+7$, that is not generally considered a "quadratic equation" in the commonly used sense. Most of the time, that is taken to mean a functional relationship between two variables, namely $y$ and $x$. Because the right hand side takes the form of a quadratic polynomial, you are justified in calling it a "quadratic function" of $x$. When writing the relationship between two variables in this form, you're looking to answer questions like: what is the value of $y$ for a given value of $x$? What does a plot of $y$ against $x$ look like? And so forth.

A quadratic equation is (already, or can easily be rearranged into) something of the canonical form $ax^2 + bx +c =0$. The last term on the left hand side and is a constant term while the right hand side is zero.

So these are quadratic equations:

1. $x^2 + x +7 =0$ (already in the canonical form)

2. $x^2 + x + 7 = 2$ (can be immediately rearranged into the canonical form)

3. $x^2 + x + 7 = k$ (where $k$ is specified as a constant, even if it's not a known constant, allowing rearrangement into the proper form)

Note that the quadratic functional relationship $y = x^2 + x +7$ can be made into a quadratic equation if we ask and try to answer questions like:

1. What value(s) of $x$ makes $y = 10$? In this case $x^2 + x +7 =10$, which is a quadratic equation with two real roots, so you have your two possible $x$ values.

2. Does the curve $y = x^2 + x +7$ intersect the $x$ axis? The answer is 'no' because the quadratic equation $x^2 + x + 7 =0$ has no real roots, only complex ones.

I hope I've understood the essence of your question and answered clearly enough.

Answer 3

It could be taken as both a relationship between $y$ and $x$ or as a quadratic equation, depending on the context of the problem.

Most of the time the form $y=ax^2+bx+c$ is used to indicate the relationship between $y$ and $x$. This is called is a quadratic function. But if it is given that $y$ is a constant, then you could simply rewrite it as $ax^2+bx+(c-y)=0$ and this is a quadratic equation.

The difference between these two is that $y$ in a quadratic function could have any value depending on the value of $x$. Whereas $y$ in a quadratic equation is independent of $x$.

Hope that clears it up.