# Quadratic Equations

An equation of the form \(Q\left( x \right) = 0\) is called a quadratic equation. The following are all examples of quadratic equations:

\[\begin{align}& 3{x^2} - 2x - 2 = 0\\&1 - 2x - 3{x^2} = 0\\&\sqrt 2 - \sqrt 3 {x^2}\;\;\, = 0 \end{align}\]

It is possible that an equation may not originally be written in the form \(Q\left( x \right) = 0\) but can be rearranged to this form. For example, the equation

\[{x^2} - 2x + 1 = x - 1\]

can be rearranged to

\[{x^2} - 3x + 2 = 0\]

and will therefore be a quadratic equation. Similarly, the equation

\[{\left( {x + 1} \right)^3} = {x^3}\]

seems to be a cubic equation (an equation in which the highest degree term is a cubic term), but it simplifies to a quadratic upon expansion of the left side:

\[\begin{align}& {x^3} + 3{x^2} + 3x + 1 = {x^3}\\&\Rightarrow \,\,\,3{x^2} + 3x + 1 = 0 \end{align}\]

The general quadratic equation is of the form

\[a{x^2} + bx + c = 0\]

**Solved Example 1:** Which of the following are quadratic expressions in the variable *x*?

- \(a + b{x^2}\)
- \(1 - \sqrt 2 {x^2} + \sqrt 7 x\)
- \(x + \frac{{{a^2}}}{x}\)
- \(1 - xy + {y^2}\)

**Solution:** The correct options are (A) and (B). In option (C), the expression is obviously not quadratic. In option (D), the expression is quadratic in the variable *y*, and *not* in the variable *x*.

**Solved Example 2:** Which of the following are quadratic equations in the variable *x *(or can be modified to quadratic equations)?

- \({\left( {1 + x} \right)^2} = {x^2}\)
- \({\left( {1 + 2x} \right)^3} - 2{x^3} = 0\)
- \(\frac{{x + 1}}{{x + 2}} = \frac{{x + 3}}{{x + 4}}\)
- \({\left( {\sqrt 2 x + \sqrt 3 } \right)^2} - {x^2} = 2x + 3\)

**Solution:** (D) is the correct option. In options (A) and (C), the equations will reduce to linear equations upon expansion and rearrangement, while in option (B), the equation will rearrange to a cubic rather than a quadratic equation. In option (D), the equation can be written as follows (verify this):

\[{x^2} + \left( {2\sqrt 6 - 2} \right)x = 0\]