Completing the Square

[Gretta99]

I'm taking a calculus class in the fall, so I decided to get a precalculus book to brush up on my math. I'm working on completing the square, and I did the problem, but it doesn't match up with the answer in the back of the book. They did it the short way, but I wanted to learn how to really complete the square. Here's what I've got, what am I doing wrong? I know that somehow I managed to flip my parabola upside down, but I'm not quite sure what I did. I'm also supposed to find the vertex, the x-intercepts, and the y-intercept.
-1/2 x^2-x+12
I divided both sides by -1/2
x^2+2x-24
x^2+2x=24
x^2+2x+1^2=24+1^2
x^2+2x+1=25
sqrt((x+1)^2)=sqrt(25)
x+1=±25
x-intercepts=-24,26
y=(0+1)^2-25
y-intercepts=-24
vertex: (-1,-25)
Any help would be greatly appreciated!

 

[mmm4444bot]

y = -1/2 x^2 - x + 12
I divided both sides by -1/2
-2y = x^2 + 2x - 24
x^2 + 2x = 24 - 2y
x^2 + 2x + 1^2 = 24 + 1^2 - 2y
x^2 + 2x + 1 = 25 - 2y

(x + 1)^2 = 25 - 2y

y - 25/2 = -(1/2)(x + 1)

y = -(1/2)(x + 1)^2 + 25/2

With the "vertex form of a quadratic equation" y = a(x + h)^2 + k, the vertex coordinates are (h,k)

After completing the square, we see that the last equation in red above is written in "vertex form".

Can you see (h, k) ?

The y-intercept occurs when x = 0. Evaluate y for x = 0: (0, ?)

The x-intercepts occur when y = 0. Solve the following to find (?, 0) and (??, 0).

(1/2)(x + 1)^2 = 25/2

Cheers, Mark

 

[tkhunny]

I'm taking a calculus class in the fall, so I decided to get a precalculus book to brush up on my math. I'm working on completing the square, and I did the problem, but it doesn't match up with the answer in the back of the book. They did it the short way, but I wanted to learn how to really complete the square. Here's what I've got, what am I doing wrong? I know that somehow I managed to flip my parabola upside down, but I'm not quite sure what I did. I'm also supposed to find the vertex, the x-intercepts, and the y-intercept.
-1/2 x^2-x+12
I divided both sides by -1/2
x^2+2x-24
x^2+2x=24
x^2+2x+1^2=24+1^2
x^2+2x+1=25
sqrt((x+1)^2)=sqrt(25)
x+1=±25
x-intercepts=-24,26
y=(0+1)^2-25
y-intercepts=-24
vertex: (-1,-25)
Any help would be greatly appreciated!

This is a little bit scary. I am glad you are brushing up.

#1) What parabola? You have an expression. "" What is that? It's not an equation. It's not a function.

#2) "I divided both sides by -1/2" -- Since you have only one "side", this is a very odd statement. Frankly, this would be legitimate ONLY if it were equivalent to zero.

-1/2 x^2-x+12 = 0

Divide by -1/2

x^2 + 2x - 24 = 0

This is most certainly NOT acceptable if it is equivalent to 'y'.

-1/2 x^2-x+12 = y

Divide by -1/2

x^2 + 2x - 24 = -2y

... or in function notation...

f(x) = -1/2 x^2-x+12

Divide by -1/2

-2f(x) = x^2 + 2x - 24 = g(x) -- It's a different function!

Let's see if you can get this far cleanly and completely before we move on.