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}

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