Question

Solving Quadratic Equations by Factoring. Solve each equation by factoring.

1) (k+1)(k-5)=0

Answer 1

This has already been factored. All we need to do now is use the zero product property



(k+1)(k-5)=0


k+1=0 or k-5=0


k=-1 or k=5

Answer 2

What about one that doesn't have a zero? xsquared-11x+19=-5

Answer 3

x^2-11x+19=-5


x^2-11x+19+5=0


x^2-11x+24=0


(x-8)(x-3)=0


x-8=0 or x-3=0


x=8 or x=3

Answer 4

I'm going to try the next one on my own using your answer as a reference. Please correct me if I did something wrong.

n^2+7n+15=5

n^2+7n+15+5=0

Is it like this so far? Or would the 5 be a negative 5?

Answer 5

on the right side, you're basically saying 0+5, so you need to subtract it from both sides

Answer 6

So, n^2+7n+10=0?

Answer 7

yep

Answer 8

Yay! Okay so,

n^2+7n+10=0
(n-5) (n-2)=0
n=5 or n=2 ?

Answer 9

close...

Answer 10

but it should be

(n+5)(n+2)=0

Answer 11

since 5+2 = 7 and 5*2 = 10

Answer 12

But the answer has to be =-5 or =-2

Answer 13

n^2+7n+10=0


(n+5)(n+2)=0


n+5=0 or n+2=0


n=-5 or n=-2

Answer 14

Ok, got it. That makes more sense.
Next one....

n^2-10n+22=-2

n^2-10n+24=0

So far, so good?

Answer 15

yep you added 2 to both sides, so far so good

Answer 16

now you need to find two numbers that satisfy the following


a) they multiply to 24 (the last number)

AND

b) they add to -10 (the second coefficient)

Answer 17

n^2-10n+24=0

(n+2)(n-12)

Like that? Or it can't be a negative cause it's positive 24?

Answer 18

well 2+(-12) = -10 (so far so good)


but...


2*(-12) = -24, which is the opposite of what we want (positive 24)

Answer 19

it turns out that these two numbers are: -6 and -4


Since -6+(-4) = -10 and (-6)*(-4) = 24

Answer 20

Wow!! I don't even know why I hadn't thought of those two numbers.

So n=-6 or n=-4

Answer 21

not quite

Answer 22

those two numbers help us factor, they aren't the answers

Answer 23

n^2-10n+24=0


(n-6)(n-4)=0 ... notice the factorization has -6 and -4 (our two numbers)


n-6=0 or n-4=0


n=6 or n=4


So the answers are n=6 or n=4

Answer 24

so the answers are the opposite of the numbers that helped us factor (since we're isolating them on the other side)

Answer 25

Alright, I think I'm getting it now but I'm going to try the next one since it has a positive answer.

n^2+3n-12=6

n^2+3n-18=0

So I use -3 and 6, right?

Answer 26

yep

Answer 27

n^2+3n-12=6

n^2+3n-18=0

(n-3)(n+6)=0

n-3=0 or n+6=0

Not sure if I did that right

Answer 28

you did that right


you need to do one more step and solve for n in each equation

Answer 29

Would it be turned to opposites? n=3 or n=-6

Answer 30

yep, you got it

Answer 31

Ok, so the next one is a little different.

6n^2-18n-18=6

Answer 32

6n^2-18n-18=6


6n^2-18n-18-6=0


6n^2-18n-24=0


6(n^2-3n-4)=0


n^2-3n-4=0/6


n^2-3n-4=0


(n-4)(n+1)=0


n-4=0 or n+1=0


n=4 or n=-1

Answer 33

n^2-3n-4=0/6

that step is being divided?

Answer 34

yes, I'm dividing both sides by 6


this is so I can effectively eliminate it and not worry about it

Answer 35

Ok, got it.
But I can't add anything to both sides on the next question.
What do I do?

7r^2-14r=-7

Answer 36

you can add 7 to both sides

Answer 37

why add 7? because on the right it becomes -7+7 which becomes 0

Answer 38

Add 7 to -14r?

Answer 39

yes, but you cannot combine the terms so you just leave them as they are

Answer 40

so the left side will be 7r^2-14r+7

Answer 41

Oh okay.

7r^2-14r+7=0

What happens on the next step?

Answer 42

now factor out the GCF 7 to get


7(r^2-2r+1)=0

Answer 43

Next step would be r^2-2r+1=0/7 ?

Answer 44

yep then you would get r^2-2r+1=0

Answer 45

So r-2=0 or r+1=0 ?

Answer 46

r^2-2r+1=0


(r-1)(r-1)=0


r-1=0 or r-1=0


r=1 or r=1


So the only answer is r=1

Answer 47

Okay, got it.

Next one:

a^2+2a-3=0

What happens if it already equals 0?

Answer 48

you then factor and use the zero product property to solve

Answer 49

This is the next page of the worksheet and the directions say to solve each equation by completing the square. Forgot to mention that

Answer 50

a^2+2a-3=0


a^2+2a=3


a^2+2a+1=3+1 ... Note: I'm adding half of 2 squared to both sides


(a+1)^2 = 4


a+1 = sqrt(4) or a+1 = -sqrt(4)


a+1= 2 or a+1 = -2


a = 1 or a = -3

Answer 51

Wow this is definitely confusing.
So the next one...

p^2+16p-22=0

p^2+16p=22

So far, so good? Not sure where to go with the next step with the half of two squared

Answer 52

take half of the second coefficient 16 to get 8


then square it to get 64


then add 64 to both sides

Answer 53

we do all this to get p^2+16p+64 on the left side


this is a perfect square and it factors to (p+8)^2

Answer 54

What happens to the 22?

Answer 55

you're also adding 64 to the right side


So add 64 to 22 to get 86

Answer 56

So p^2+16p+64=86?

Answer 57

yep

Answer 58

What is the next step?

Answer 59

now factor the left side

Answer 60

So factoring the left side gives us


p^2+16p+64


(p+8)(p+8)


(p+8)^2



So we now have (p+8)^2 = 86

Answer 61

So then p+8=sqrt86?

Answer 62

or p+8 = -sqrt(86), remember the plus/minus

Answer 63

So then p+8=sqrt86?

Answer 64

Yup, already written down

Answer 65

oh ok

Answer 66

So then p+8=sqrt86?

Answer 67

so p+8 = 9.3 or p+8=-9.3 ?

Answer 68

approximately yes

Answer 69

Ok. Next page the directions says to solve each equation with the quadratic formula.

m^2-5m-14=0

Answer 70

The Quadratic formula is


m = (-B +- sqrt( B^2-4AC ))/(2A)



In the case of m^2-5m-14, A=1, B=-5, and C=-14



m = (-B +- sqrt( B^2-4AC ))/(2A)



m = (-(-5) +- sqrt( (-5)^2-4(1)(-14) ))/(2*1)



m = (5 +- sqrt( (-5)^2-4(1)(-14) ))/(2*1)



m = (5 +- sqrt( 25+56 ))/(2)



m = (5 +- sqrt( 81 ))/(2)



m = (5 +- 9)/(2)



m = (5 + 9)/(2) or m = (5 - 9)/(2)



m = (14)/(2) or m = (-4)/(2)



m = 7 or m = -2



So the solutions are m = 7 or m = -2

Answer 71

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Answer 72

yeah that's probably a good idea lol


I know a bit about him, but probably not anything in great depth. What do you want to know?