Answer to Question #186787 in Algebra for Jethro

Question

Complete the square of the terms inside the square brackets:

ax² + bx + c = a [x2 + ( b/a) x + c/a]

Answer 1

$ax^2+bx+c=a\Big[x^2+\Big(\dfrac{b}{a}\Big)x+\dfrac{c}{a}\Big]$

$=a\Big[x^2+\Big(\dfrac{b}{a}\Big)x+\dfrac{c}{a}+\dfrac{b^2}{4a^2}-\dfrac{b^2}{4a^2}\Big]$

$=a\Big[\Big(x^2+\Big(\dfrac{b}{a}\Big)x+\dfrac{b^2}{4a^2}\Big)-\dfrac{b^2}{4a^2}+\dfrac{c}{a}\Big]$

$=a\Big[\Big(x+\dfrac{b}{2a}\Big)^2-\dfrac{b^2}{4a^2}+\dfrac{c}{a}\Big]$

$=a\Big[x+\dfrac{b}{2a}\Big]^2-\dfrac{b^2}{4a}+c$