Question

Which expression is the general transformation obtained by completing the square?

Options

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

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

• \[ax^2+bx+c=\left(x+\frac{b}{2a}\right)^2+\frac{c}{a}\]

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

Answer 1

First factor out \[a\] from the quadratic expression. Completing the square then gives the shift \[x+\frac{b}{2a}\] and the remaining constant \[\frac{c}{a}-\frac{b^2}{4a^2}\].