Completing the Square Flashcards

All 6 cards in this deck

What is the completed square form of $x^2 + bx + c$?

$\left(x+\frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$
e.g. $x^2+6x+1 = (x+3)^2 - 9 + 1 = (x+3)^2 - 8$

What are the steps to write $ax^2+bx+c$ in the form $k(x+m)^2+n$?

e.g. $2x^2 - 16x + 7$

1. Factor out the coefficient of $x^2$ from the first two terms: $2(x^2-8x)+7$
2. Complete the square inside: $2[(x-4)^2-16]+7$
3. Expand and simplify: $2(x-4)^2 - 25$

What does $k(x+m)^2+n$ expand to?

$kx^2 + 2kmx + km^2 + n$

For $k(x+m)^2+n$ with $k>0$, what is the minimum value and where does it occur?

Minimum value $n$, occurring at $x=-m$.
e.g. $2(x-4)^2-25$ has minimum $-25$ at $x=4$

For $k(x+m)^2+n$ with $k<0$, what is the maximum value and where does it occur?

Maximum value $n$, occurring at $x=-m$.
e.g. $-3(x+2)^2+7$ has maximum $7$ at $x=-2$

True or false? The minimum of $(x-3)^2+5$ is $5$ and it occurs at $x=-3$.

False. The minimum is $5$ but it occurs at $x=3$.