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?

1.  $$a + b{x^2}$$
2.  $$1 - \sqrt 2 {x^2} + \sqrt 7 x$$
3.  $$x + \frac{{{a^2}}}{x}$$
4.  $$1 - xy + {y^2}$$

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)?

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

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

grade 10 | Questions Set 2
grade 10 | Questions Set 1