Completing the square; for turning points.

Hi guys;

I have been studying "completing the square" in relation to Algebraic functions and graphs. I have learned that using "completing the square" will give you the turning point.

With x2 + 4x + 1 = 0

When I completed the square; I got the result of (x+2)²=3

Now, apparently; It gives us the vertex (turning point) of x2 + 4x + 1: (-2, -3).

However I am somewhat at a loss at how exactly those figures were derived.

Can someone please help please?

Reply 1

Kvothe the Arcane

20

Original post by apronedsamurai

Hi guys;

I have been studying "completing the square" in relation to Algebraic functions and graphs. I have learned that using "completing the square" will give you the turning point.

With x2 + 4x + 1 = 0

When I completed the square; I got the result of (x+2)²=3

Now, apparently; It gives us the vertex (turning point) of x2 + 4x + 1: (-2, -3).

However I am somewhat at a loss at how exactly those figures were derived.

Can someone please help please?

Let's say a quadratic is generally

(x^2)=b

.

(x-a)^2=b

is just a

f(x-a)

transformation. It's shifted to the right by a units.
So the stationary point (0,b) becomes

(a,b)

. The

-b

is just the equivalent y point as making the bracket 0 means that y=-b

(edited 9 years ago)

Reply 2

User947387

OP

17

Original post by keromedic

I think you mean (-2,3)

No, -3 was the value given on the tutorial website.

Reply 3

TenOfThem

18

Original post by apronedsamurai

With x2 + 4x + 1 = 0

When I completed the square; I got the result of (x+2)²=3

Now, apparently; It gives us the vertex (turning point) of x2 + 4x + 1: (-2, -3).

x^2 + 4x + 1 = (x+2)^2 - 3

do not rearrange

you should realise that

(x+2)^2

is 0 or positive

Therefore the whole thing is smallest when (x+2) = 0

So

(-2,-3)

Reply 4

User947387

OP

17

So sorry; still not quite getting it.

I understand (x+2)=0; therefore; x=2. But what of -3? Is this what we call substitution; and this is the method used always?

Reply 5

TenOfThem

18

Original post by apronedsamurai

So sorry; still not quite getting it.

I understand (x+2)=0; therefore; x=2. But what of -3? Is this what we call substitution; and this is the method used always?

You had

y = (x+2)^2 - 3

What does that equal when the bracket is zero

Reply 6

SuperiorGenius

Why have you made it =3? Move that 3 back to the other side before you get confused. The original equation was equal to 0....

Reply 7

Zen-Ali

8

Original post by apronedsamurai

...

It's fairly straightforward.

We know that

(x + 2)^2 \geq 0

for

x \in \mathbb{R}

because if that bracket becomes negative, squaring it will make it positive and squaring that bracket if it was positive would make it positive: this means that the minimum must occur when