1. The height of water shooting from a fountain is modeled by the function f(x) = −4x2 + 24x − 29 where x is the distance from the spout in feet. Complete the square to determine the maximum height of the path of the water. (2 points)

−4(x − 3)^2 − 29; The maximum height of the water is 3 feet.
−4(x − 3)^2 − 29; The maximum height of the water is 29 feet.
−4(x − 3)^2 + 7; The maximum height of the water is 7 feet.
−4(x − 3)^2 + 7; The maximum height of the water is 3 feet.

2. In the function f(x) = 4(x2 − 6x + [blank] ) + 20, what number belongs in the blank to complete the square?

Question

1. The height of water shooting from a fountain is modeled by the function f(x) = −4x2 + 24x − 29 where x is the distance from the spout in feet. Complete the square to determine the maximum height of the path of the water. (2 points)
 	
	−4(x − 3)^2 − 29; The maximum height of the water is 3 feet.
	−4(x − 3)^2 − 29; The maximum height of the water is 29 feet.
	−4(x − 3)^2 + 7; The maximum height of the water is 7 feet.
	−4(x − 3)^2 + 7; The maximum height of the water is 3 feet.

2. In the function f(x) = 4(x2 − 6x + [blank] ) + 20, what number belongs in the blank to complete the square?

iwillrektyou · Answer

For #2, I think the blank is 9, for #1, I'm totally confused.

jhannybean · Answer

$$f(x)=-4x^2+24x-29$$ in the method of completing the square, this is how I would go about solving it. 

first group your x-terms together. $(-4x^2+24x)-29$

Now out of the grouped x-terms, factor out the LCM, -4.: $-4(x^2-6x)-29$

We complete the square inside the parenthesis. therefore we need to find a new "c" value. $c= \left(\dfrac{-6}{2}\right)^2 = (-3)^2 = 9$ 

plug this back in to the parenthesis to complete the quadratic form, $ax^2+bx+c$
$-4(x^2-6x+9)-29$

Now this is the problem child, a lot of people forget that whatever you do inside the ( ), you `must also do outside the ( )`. Because we added `+9` inside the parenthesis, we will have to `subtract` the equivalent. That is, $(-4 \cdot 9) = -36$$$-4(x^2-6x+9)-29-(-36)$$$$-4(x^2-6x+9)-29+36$$$$-4(x^2-6x+9)+7$$ Now we must simplify the stuff in ( ) because it is a perfect square after all. $$\color{red}{\boxed{-4(x-3)^2+7}}$$

iwillrektyou · Answer

Thank you so much! Can you please check if 9 is the answer to #2?

iwillrektyou · Answer

And for the first problem, the vertex is (3, 7), so is the maximum height 3 feet?

jhannybean · Answer

The maximum `height` in correlated to the `y-value`

iwillrektyou · Answer

So it is 7?

jhannybean · Answer

Yes

iwillrektyou · Answer

Thank you very much.

jhannybean · Answer

Do you understand how I went about solving it? @iwillrektyou

iwillrektyou · Answer

Yes, I also did it on my paper and got the same answer.
For #2, I got 9, using the middle coefficient. 6/2 = 3 ; 3^2 = 9

jhannybean · Answer

yes that is correct. 

(-6/2)^2 = (-3)^2 = 9 $\checkmark$

iwillrektyou · Answer

Would you mind helping me with a few more?

jhannybean · Answer

i've actually got to head off and work on my own homework, so im sorry about that!

iwillrektyou · Answer

Thank you so much for your time and patience!

jhannybean · Answer

No problem :)