Why 9/4 in Quadratic Function SOLVED

In summary, the conversation discusses solving a quadratic function and the use of completing the square to make it easier. The solution is already given in the book, but the questioner is confused about the insertion of 9/4 in the third line. The experts explain that it is used to make the equation factorable and balance the equation. They also provide an alternative method of solving the problem and suggest using LaTeX to type out mathematical equations.
  • #1
viet_jon
131
0
[SOLVED] Quadratic Function

Homework Statement

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

Homework Equations


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?
 
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  • #2
What they've done is simply completed the square. If you have an equation of the form: [itex] x^2+2ax [/itex], you can write it as: [itex] x^2+2ax = (x+a)^2 - a^2 [/itex].
 
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  • #3
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
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
Well here's how I would have done that problem. You're given:

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

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

[tex] f(x) = -3\left(\left(x-\frac{3}{2}\right)^2 - \frac{9}{4} \right) + \frac{1}{4} [/tex]
[tex] f(x) = -3\left(x - \frac{3}{2}\right)^2 + \frac{27}{4} + \frac{1}{4} [/tex]
[tex] f(x) = -3\left(x - \frac{3}{2}\right)^2 + 7 [/tex]
 
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  • #6
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?
 
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  • #8
test:

[tex]x=-3 x^2+9x+1/4[/tex]

[tex]fx=(-3)(x^2┤-3 x)+1/4\\[/tex]

[tex]fx=(-3)(x^2┤-3 x+9/4)+1/4+27/4\\[/tex]

[tex]fx=(-3)(x^2┤-3 x+9/4)+28/4\\[/tex]

[tex]fx=(-3) (x┤-3/2)^2+7\\[/tex]
 
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  • #9
viet_jon said:
test:

<tex>x^2</tex>

<tex>(x)=-3 x^2+9x+1/4</tex>
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

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]
 
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1. Why is 9/4 in Quadratic Function significant?

The number 9/4, also known as 2.25, is significant in Quadratic Functions because it represents the value of the coefficient of the squared term in the standard form of a quadratic equation. This coefficient, also known as the "a" value, determines the shape and direction of the parabola in the graph of the quadratic function.

2. How is 9/4 solved in Quadratic Functions?

To solve for 9/4 in Quadratic Functions, we can use the standard form of a quadratic equation, which is y = ax^2 + bx + c. The coefficient of the squared term, or "a" value, is equal to 9/4. We can then plug this value into the quadratic formula or use other methods such as factoring or completing the square to solve for the roots of the equation.

3. Can 9/4 be simplified in Quadratic Functions?

Yes, 9/4 can be simplified in Quadratic Functions. In fact, it is common practice to simplify fractions in quadratic equations to make them easier to work with. In this case, 9/4 can be simplified to 2.25, which is a decimal value.

4. What role does 9/4 play in the graph of a Quadratic Function?

The number 9/4, or 2.25, plays a crucial role in the graph of a Quadratic Function. As the coefficient of the squared term, it determines the direction and shape of the parabola. If the value of 9/4 is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. The value also affects the position of the vertex, which is the highest or lowest point on the graph.

5. Are there any real-life applications of 9/4 in Quadratic Functions?

Yes, there are several real-life applications of 9/4 in Quadratic Functions. For example, it can be used to calculate the maximum height of a projectile in physics or to determine the optimal production level for a business in economics. It is also used in engineering to design and optimize structures such as bridges and buildings.

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