1. Nov 28, 2007

### viet_jon

1. The problem statement, all variables and given/known data

solve (f(x)=-3 x^2 + 9 x + 1/4 )
fx=(-3)(x^2-3 x)+1/4

fx=(-3)(x^2-3 x+9/4)+1/4+27/4

fx=(-3)(x^2-3 x+9/4)+28/4
fx=(-3) (x-3/2)^2+7

2. Relevant equations

3. The attempt at a solution

the solution is already given in the book, but I don't understand why 9/4 was inserted in the third line (bolded). I know at the end of the equation, 27/4 is added to balance out the 9/4.......but why put in 9/4 there?

2. Nov 28, 2007

### Kurdt

Staff Emeritus
What they've done is simply completed the square. If you have an equation of the form: $x^2+2ax$, you can write it as: $x^2+2ax = (x+a)^2 - a^2$.

Last edited: Nov 28, 2007
3. Nov 28, 2007

### rock.freak667

f(x)=-3 x^2 + 9 x + 1/4

they just put 27/4 and -27/4 to make completion easier

f(x)=-3 x^2 + 9 x -27/4+ 1/4 +27/4
=-3(x^2-3x+9/4)+7
=-3(x-3x/2)^2 +7

expand (x-3x/2)^2 and check it yourself

4. Nov 28, 2007

### viet_jon

yea bro...

the thing is, how do you come up with 9/4 from to make it easier?

I'm on purplemath.com right now, trying to figure it out without bothering you guys... hopefully I can figure this one out.

5. Nov 28, 2007

### Kurdt

Staff Emeritus
Well heres how I would have done that problem. You're given:

$$f(x) = -3x^2 + 9x + \frac{1}{4}$$
$$f(x) = -3( x^2 -3x) + 1/4$$

Now we notice that the term in brackets is of the form $x^2+2ax$, with $a=\frac{-3}{2}$, and so we complete the square $x^2+2ax = (x+a)^2 - a^2$:

$$f(x) = -3\left(\left(x-\frac{3}{2}\right)^2 - \frac{9}{4} \right) + \frac{1}{4}$$
$$f(x) = -3\left(x - \frac{3}{2}\right)^2 + \frac{27}{4} + \frac{1}{4}$$
$$f(x) = -3\left(x - \frac{3}{2}\right)^2 + 7$$

Last edited: Nov 28, 2007
6. Nov 28, 2007

### viet_jon

okie.........i got it.

so the point of adding -9/4 inside the brackets is so that you can make it factorable. And since you added a (-), now you have to minus it to even it out.

thnkx.....can't believe it took me so long.

anyhow, how do you type the math out like that?

Last edited: Nov 28, 2007
7. Nov 29, 2007

### jaime2000

8. Nov 29, 2007

### viet_jon

test:

$$x=-3 x^2+9x+1/4$$

$$fx=(-3)(x^2┤-3 x)+1/4\\$$

$$fx=(-3)(x^2┤-3 x+9/4)+1/4+27/4\\$$

$$fx=(-3)(x^2┤-3 x+9/4)+28/4\\$$

$$fx=(-3) (x┤-3/2)^2+7\\$$

Last edited: Nov 29, 2007
9. Nov 29, 2007

### Kurdt

Staff Emeritus
If you want to test then there are plenty of preview websites. Here is one of them.

http://at.org/~cola/tex2img/index.php [Broken]

Last edited by a moderator: May 3, 2017