Quadratic Expressions

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 \(y\):

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

  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school

0