Question

What is the general form of a quadratic equation?

Answer 1

Hint: Since, algebra is a topic that covers an essential part of mathematics. In that part, we use to read a chapter that is based on Theory of expression. In this chapter of algebra, we read about a quadratic equation in which that expression has two roots. That satisfies that quadratic equation. Here is the quadratic equation as:
$a{{x}^{2}}+bx+c=0$
Where, $a$ , $b$ and $c$ are constants.
Since, it is a function, we can also write it as:
$f\left( x \right)=a{{x}^{2}}+{{a}_{2}}x+{{a}_{3}}$
Where, ${{a}_{1}}$ , ${{a}_{2}}$ and ${{a}_{3}}$ are constants.

Complete step by step solution:
Let us start from the definition. As we know that quadratic equation is a function and a function is a mathematical expression of variables in the form of degree. So, we will try to understand which type of function is the quadratic equation. Here, we can understand it as:
A mathematical expression that is in the form of a variable has maximum degree of two and three constants.
Since, a mathematical expression would be like this:
$\Rightarrow {{a}_{1}}{{x}^{n}}+{{a}_{2}}{{x}^{n-1}}+{{a}_{3}}{{x}^{n-2}}+...+{{a}_{n+1}}$
Where, ${{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n+1}}$ are constants.
So, for a quadratic equation, we will put $2$ in the place of $n$ . So we have mathematical expression as:
$\Rightarrow {{a}_{1}}{{x}^{2}}+{{a}_{2}}{{x}^{2-1}}+{{a}_{3}}{{x}^{2-2}}$
Now, we will do necessary calculation to get the quadratic equation as:
$\Rightarrow {{a}_{1}}{{x}^{2}}+{{a}_{2}}{{x}^{1}}+{{a}_{3}}{{x}^{2}}$
Since, if the power of any variable is zero, then that variable is equal to one i.e. ${{x}^{0}}=1$ , so we will put this value in the above equation as:
$\Rightarrow a{{x}^{2}}+{{a}_{2}}x+{{a}_{3}}\times 1$
Now, we will get the proper quadratic equation as:
$\Rightarrow a{{x}^{2}}+{{a}_{2}}x+{{a}_{3}}$
Since, we denote a function with symbol $f\left( x \right)$ , so we can express the above equation as:
$\Rightarrow f\left( x \right)=a{{x}^{2}}+{{a}_{2}}x+{{a}_{3}}$
Where, ${{a}_{1}}$ , ${{a}_{2}}$ and ${{a}_{3}}$ are constants and no one is equal to zero.
And $f\left( x \right)$ denotes that this is a function with variable $x$ .

Note: We need to take care of some points that is about the quadratic equation that are shown below as:
Every quadratic equation has two roots.
These roots might be same, different or opposite in symbol and we can check it by using the formula:
${{a}_{2}}^{2}-4{{a}_{1}}{{a}_{3}}$ .
These roots satisfy the quadratic equation if we apply them in the equation.