Question

Fay threw a basketball from the basketball court toward the hoop. The quadratic equation that models the height of the ball after t seconds is(hx)=−16t2 +20t+6.if the hoop is 10 feet high, how long is the
ball in the air before the ball goes through the hoop?

Answer 1

@ganeshie8

Answer 2

set the height function equal to 10 and solve t

Answer 3

How do we solve for t

Answer 4

start by setting the function equal to 10

Answer 5

10=-16t^2+20t+6

Answer 6

Like that?

Answer 7

yes subtract 10 both sides

Answer 8

and maybe cancel any common constant factors

Answer 9

you will get a quadratic

Answer 10

you can solve a quadratic by factoring or by using quadratic formula

Answer 11

-16t^2+20t=4

Answer 12

\[10=-16t^2+20t+6\]

subtracting 10 both sides gives
\[0=-16t^2+20t-4\]

right ?

Answer 13

Yes

Answer 14

cancel -4 through out, you get

\[4t^2-5t+1 = 0\]

Answer 15

Why do we cancel 4

Answer 16

Then how do we factor it

Answer 17

you're allowed to divide a non zero number both sides

Answer 18

dividing -4 through out gives you small numbers

Answer 19

Oh ok

Answer 20

small numbers are easy to handle

Answer 21

So then how do we factor it

Answer 22

\[4t^2-5t+1 = 0 \]

pick two numbers such that :
1) they add up to -5
2) they multiply to 4

Answer 23

I mean -1 -4

Answer 24

yes! write the middle term using them
\[4t^2\color{Red}{-5t}+1=0\]

Answer 25

\[4t^2\color{Red}{-1t - 4t}+1=0\]

Answer 26

factor the GCF from first two terms and the last two terms

Answer 27

I never took this method can't. We use quadratic formula

Answer 28

okay use quadratic formula then

Answer 29

Do we use the simplest factor

Answer 30

4t^2-5t+1

Answer 31

yes

Answer 32

work it fast, otherwise @Jhannybean going to forcibly teach you completing the square !

Answer 33

Hahaha ok

Answer 34

I got 5+-sqrt5 /8

Answer 35

doesn't look correct, try again

Answer 36

\[4t^2-5t+1 = 0\]\[t^2 -5t+4=0\]\[(t-4)(t-1)=0\] Since we multiplied the 4 over to the one to make it a quadratic equation that's easier to solve, we have to divide by the coefficient as well. \[\left(t-\frac{4}{4}\right)\left(t-\frac{1}{4}\right)=0\] simplify the fractions you can, and for those that you cannot, multiply the denominator to the variable. \[(t-1)(4t-1)=0\]

Answer 37

Soory I didn't understand this method

Answer 38

:(

Answer 39

you're taking a complex quadratic equation and simplifying it in order to find your factoredform

Answer 40

I still don't learn how to simplify quadratic equations that's why

Answer 41

Then I'll teach you this method of quick factorization :)

Answer 42

Now?

Answer 43

Let's work it step by step. you understand how you found the \[4t^2 -5t-1=0\] right?

Answer 44

No

Answer 45

So you did not follow what @ganeshie8 was guiding you through?

Answer 46

Wait let we go back and check

Answer 47

Take a minute to review the stuff up there ^ when you're done looking through it, let me know.

Answer 48

i did simplify it in another way
4(4t^2+5t-1)=0
4 will be canceled

Answer 49

then we are left with 4t^2+5t-1

Answer 50

@Jhannybean

Answer 51

but then i don't know how to simplify more

Answer 52

Okay, do you do understand how to get to this point.

Answer 53

yess

Answer 54

I find it really hard to factor out a quadratic function that has a coefficient in the front, in this case a 4.

Answer 55

so is there another way

Answer 56

So what I do is this simple trick, but you have to constantly remember that you are modifying the equation to make it easier to solve, but the function WILL NOT CHANGE.

Answer 57

ok

Answer 58

This way that I am teaching you is probably the easiest method of going about factoring out a quadratic equation without having to use the quadratic formula

Answer 59

yay im so happy

Answer 60

So in order to make it easier to factor, I simply took the 4 that was my coefficient and multiplied it t the constant. This made it easier to find two numbers that add to give me -5, and multiply to give me +4

Answer 61

\[t^2 -5t+4=0\]That is how I got to this step. following me so far?

Answer 62

1 min

Answer 63

so we divide the coefficient with 4 and times the last number by 4

Answer 64

but then why did the signs change

Answer 65

In that step we are simply 'creating' an easier quadratic function to factor.Would you agree this: \[4t^2 -5t+1\] is a lot harder to factor than \[t^2-5t+4\]

Answer 66

Just follow me step by step, do not jump to conclusions just yet.

Answer 67

ok
so signs didn't change

Answer 68

No, they didn't.

Answer 69

oh ok

Answer 70

So now you understand how we came to \(t^2-5t+4\), yes?

Answer 71

yes

Answer 72

Okay.

Answer 73

Now we find two numbers that ADD to give -5, but MULTIPLY to give -4. What can those be?

Answer 74

but wait why -4

Answer 75

u mean 4

Answer 76

so its -1 -4

Answer 77

yes,typo,sorry. Glad you caught that

Answer 78

yes, -4 , -1.

Answer 79

ok wait before we continue if we dot have a leading coefficient we divide by and mutually by what

Answer 80

dont*

Answer 81

If we don't have a leadng coefficient, then we just factor the quadratic like normal

Answer 82

The only reason we are adding on a few extra steps like simplifying the quadratic is just to make it easier to factor.

Answer 83

But moving on, we will have \((t-1)(t-4)\) , but if you expand this, you will only have the simplified quadratic function, \(t^2-5t+4\)

Answer 84

so this method is for if we have leading coefficient

Answer 85

Yes! :) There you go.

Answer 86

:) ok lets continue

Answer 87

This is where it might get a little tricky, so pay close attention.

Answer 88

ok

Answer 89

Like I mentioned before, we need to use this simplified form to revert back to our original equation: \(4t^2-5t+1\). Because we initially multiplied the 4 to make it easier to factor, we need to now divide by the same coefficient so we can get back the original function

Answer 90

Thats why from \[(t-1)(t-4)\] we will divide both of those values by 4.\[\left(t-\frac{1}{4}\right)\left(t-\frac{4}{4}\right)\]

Answer 91

ok

Answer 92

so its 1/4 the ANSWER

Answer 93

No not yet.

Answer 94

OH OK

Answer 95

After dividing, we cam simplify the fractions, whichever fractions simplify. \(\frac{4}{4} = 1\) and \(\frac{1}{4}\) stays the same.

Answer 96

oh ok

Answer 97

So we will have \[\left(t-\frac{1}{4}\right)\left(t-1\right)\]