Quadratic Expressions

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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 \(y\):

\[\begin{align}&{Q_1}\left( x \right)\,:\;2{x^2} - 5x + 2\,\\&{Q_2}\left( y \right)\,:\;1 - 7{y^2}\end{align}\]