The polynomial \(a{x^2} + bx + c\) where \(a  \ne 0, a, b\) and \(c\) are real numbers, is called a quadratic expression and when a quadratic expression is equal to 0, it is known as a quadratic equation. Let the quadratic expression is represented by:  

\[y = a{x^2} + bx + c = a\left( {{x^2} + \frac{b}{a}x + \frac{c}{a}} \right)\] \[=a\left\{ {{{\left( {x + \frac{b}{{2a}}} \right)}^2} - \frac{{{b^2}}}{{4{a^2}}} + \frac{c}{a}} \right\}\] \[ = a\left\{ {{{\left( {x + \frac{b}{{2a}}} \right)}^2} + \left( {\frac{{4ac - {b^2}}}{{4{a^2}}}} \right)} \right\}\]

Example 6: Find the minimum value of the expression \(5{x^2} - 10x - 5\)

Solution: The expression can be written as:

\(y = 5({x^2} - 2x - 1)\)

Now complete the square in the bracket:

\(y = 5\left[ {{{(x - 1)}^2} - 2} \right]\) or \(y = 5{(x - 1)^2} - 10\) 

We are trying to find the minimum value that this graph can be. \({(x - 1)^2}\) must be zero or positive, since squaring a number always gives a positive answer. So the minimum value will occur when 

\({(x - 1)^2} = 0\),  which is when \(x = 1\).

Alternate solution: by using the formula \(\left[ {\frac{{4ac - {b^2}}}{{4a}}} \right]\), minimum value of the expression is \(\left[ {\frac{{ - 4 \times 5 \times 5 - 100}}{{4.5}}} \right] = - 10\)