Quadratic Expressions by Cuemath
A quadratic expression is a polynomial with degree two. Some examples of quadratic expressions:
\[1  x + {x^2},\,\,\,\,\,\,{y^2} + 1,\,\,\,\,\,\,  3{z^2}\]
Some examples of expressions which are not quadratic:
\[  x + 2,\,\,\,\,3 + {y^3}  {y^2} + 1,\,\,\,\,{z^{10}}  3\]
Note that for a polynomial expression to classify as a quadratic, the powers of the coefficients of the variables are irrelevant. Thus, the following are also quadratic expressions:
\[  \sqrt 2 {x^2} + \sqrt 3 x  \frac{1}{2},\,\,\,\,{\pi ^3}{y^2}  {2^{  \frac{1}{3}}}y + 1,\,\,\,\,\frac{{{z^2}}}{{{3^{10}}}}  \frac{z}{{{3^5}}}\,\]
We can use any letter to represent the variable in a quadratic expression. For example, the expression \(  3{t^2} + t  2\) is a quadratic in the variable \(t\). The expression \(x  a  {a^2}\) is not a quadratic in \(x\), but it is a quadratic in \(a\).
If the variable is \(x\), then the simplest quadratic expression is \({x^2}\), whereas the general quadratic expression is of the form \(a{x^2} + bx + c\). There can be at the most three terms in a quadratic expression:

the square term (the term containing \({x^2}\))

the linear term (the term containing \(x\))

the constant term (the term which is independent of the variable)
We will use the notation \(Q\left( x \right)\) to represent a quadratic expression which is quadratic in the variable \(x\). A quadratic in \(y\) can then be represented as \(Q\left( y \right)\), and so on. To represent two different quadratics, we can use the notation \({Q_1},\,\,{Q_2}\) etc. Below are two different quadratics, one in \(x\) and one in \(y\):
\[\begin{align}&{Q_1}\left( x \right)\,:\;2{x^2}  5x + 2\,\\&{Q_2}\left( y \right)\,:\;1  7{y^2}\end{align}\]