Completing the Square Flashcards

All 11 cards in this deck

What are the steps to complete the square when the coefficient of $x^2$ is not 1?
e.g. $2x^2-12x+7$

1. Factorise the coefficient of $x^2$ out of the $x$ terms: $2(x^2-6x)+7$
2. Complete the square inside the bracket: $2[(x-3)^2-9]+7$
3. Multiply out and simplify: $2(x-3)^2-11$

What are the coordinates of the turning point of $y=(x+a)^2+b$?
e.g. $y=(x-4)^2-13$

$(-a,\;b)$
e.g. $(4,\,-13)$

For $y=a(x+p)^2+q$, is $q$ the minimum or the maximum value of $y$, and at what value of $x$ does it occur?
e.g. $y=-3(x-2)^2+7$

Minimum if $a>0$, maximum if $a<0$; either way it occurs at $x=-p$.
e.g. maximum value $7$ at $x=2$

True or false? The graph of $y=(x+5)^2-2$ has line of symmetry $x=5$.

False. The line of symmetry is $x=-5$, through the turning point $(-5,-2)$.

How do you use the completed square form to show that a quadratic expression is always positive?
e.g. $x^2+6x+11=(x+3)^2+2$

Show the constant after the squared bracket is positive: since $(x+3)^2\ge 0$, the expression is $\ge 2$, so it is always positive.

What are the steps to write a quadratic of the form $x^2+bx+c$ in completed-square form $(x+a)^2+k$?

e.g. $x^2+6x-7$

1. Halve the coefficient of $x$ to get the bracket: $(x+3)^2$
2. Subtract the square of that number: $(x+3)^2-9$
3. Add the constant term $c$: $(x+3)^2-9-7=(x+3)^2-16$

True or false? $x^2+10x+3=(x+5)^2+3$

False. You must subtract $5^2$: $x^2+10x+3=(x+5)^2-22$.

From the completed-square form of a quadratic, how do you tell whether it has a minimum or a maximum value?

e.g. $y=-3(x-2)^2+7$

Minimum if the coefficient of the squared bracket is positive, maximum if it is negative.
Here the coefficient is $-3$, so $y$ has a maximum value of $7$.

True or false? For $y=2(x-3)^2-11$, the minimum value of $y$ is $3$.

False. The minimum value of $y$ is $-11$; it occurs at $x=3$.

What does the completed-square form $(x+a)^2+k$ with $k>0$ tell you about the expression?

It is always positive, because $(x+a)^2\ge 0$ for all $x$, so the expression is at least $k$.
e.g. $(x+3)^2+2\ge 2$

True or false? $(x-5)^2-4$ is positive for all values of $x$.

False. Its minimum value is $-4$, so it is negative for some values of $x$.