Mathematico | Mathematico

Express

5 x^{2} - 2 x + 1

in the form

\frac{(a x + b)^{2} + c}{5}

where $a, b, c \in Z$

Give the value of $a b c$

Hint

After completing the square, you should see the expression

{(x - \frac{1}{5})}^{2}

If you collect the contents of the bracket into a single fraction, you get

{(\frac{5 x - 1}{5})}^{2}

Solution

We begin by completing the square:

\begin{aligned} 5 x^{2} - 2 x + 1 \\ = & 5 [x^{2} - \frac{2}{5} x] + 1 \\ = & 5 [{(x - \frac{1}{5})}^{2} - \frac{1}{25}] + 1 \\ = & 5 {(x - \frac{1}{5})}^{2} - \frac{1}{5} + 1 \\ = & 5 {(\frac{5 x - 1}{5})}^{2} + \frac{4}{5} \\ = & \frac{(5 x - 1)^{2}}{5} + \frac{4}{5} \\ = & \frac{(5 x - 1)^{2} + 4}{5} \end{aligned}