Question

can someone help explain how to solve quadratic equations by completing the square? youtube videos aren't really helping me

Answer 1

Do you have an example you want to work with?

Answer 2

h(t)=-16t^2+576t

Answer 3

Ok for this example, I'm going to use x in place of t.

So instead of h(t)=-16t^2+576t, I'm going to use h(x)=-16x^2+576x

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To complete the square for h(x)=-16x^2+576x, we can find the vertex to help us out

We do this by finding the axis of symmetry first

-16x^2+576x is in the form ax^2 + bx + c where

a = -16
b = 576
c = 0

Plug in a = -16 and b = 576 into the axis of symmetry formula below

x = -b/(2a)

x = -576/(2*(-16))

x = -576/(-32)

x = 18

So the x coordinate of the vertex is 18

Plug this into the function to get the y coordinate of the vertex

y = -16x^2+576x

y = -16(18)^2+576(18)

y = -16(324)+576(18)

y = -5184+10368

y = 5184

This is the y coordinate of the vertex.

Since the vertex is (18, 5184), we know that

(h,k) = (18, 5184)

which means

h = 18
k = 5184

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So you plug a = -16, h = 18 and k = 5184 into the general vertex equation y = a(x-h)^2 + k to get

y = a(x-h)^2 + k

y = -16(x-18)^2 + 5184

Answer 4

It's a lot I know, but once you get the hang of it, it's not too bad.

In summary, after completing the square

y = -16x^2+576x

transforms into

y = -16(x-18)^2 + 5184

Answer 5

Let me know when you're ready for the next step

Answer 6

this is the best response i've gotten so far and now i actually know how to do it! thank you so much! i am ready for the next step! :)

Answer 7

ok great

we want to find the roots of y = -16x^2+576x, which is basically the same thing as saying "solve -16x^2+576x = 0"

because y = -16(x-18)^2 + 5184 is equivalent to y = -16x^2+576x, we can solve the equation below

-16(x-18)^2 + 5184 = 0

to get the same answers

Answer 8

So let's solve -16(x-18)^2 + 5184 = 0 for x. We do so by following PEMDAS in reverse to isolate x and get it on its own side

\[\Large -16(x-18)^2 + 5184 = 0\]

\[\Large -16(x-18)^2 = -5184\]

\[\Large (x-18)^2 = \frac{-5184}{-16}\]

\[\Large (x-18)^2 = 324\]

\[\Large x-18 = \pm\sqrt{324}\]

\[\Large x-18 = \sqrt{324} \ \text{ or } \ x-18 = -\sqrt{324}\]

\[\Large x-18 = 18 \ \text{ or } \ x-18 = -18\]

\[\Large x = 18+18 \ \text{ or } \ x = -18+18\]

\[\Large x = 36 \ \text{ or } \ x = 0\]

Answer 9

So the two solutions of \(\Large -16(x-18)^2 + 5184 = 0\) are...


\[\Large x = 36 \ \text{ or } \ x = 0\]


That means the two solutions of \(\Large -16x^2+576x = 0\) are...


\[\Large x = 36 \ \text{ or } \ x = 0\]


since the two equations are equivalent

Answer 10

THANK YOU SO MUCH but i understand it all besides what the question is really asking for... do you mind if i ask you?

Answer 11

Sure go ahead

Answer 12

a projectile is thrown upward so that its distance above the ground after t seconds is given by the function h(t)=-16t^2+576t. after how many seconds does the projectile take to reach its maximum height?

Answer 13

Ok they want the x coordinate of the vertex essentially. In other words, they want the value of t when the projectile is at the max height.

Remember how I replaced t with x (I found it easier that way). The x coordinate of the vertex is x = 18.

So t = 18 is the time when the projectile is at the max height (ie the vertex)

So that's why the answer is t = 18

Answer 14

THANK YOU SO MUCH @jim_thompson5910