Solve Equations by Factoring?

Question

can u help me solve these problems and show how u get it step by step.

1. b^2=10b-16

2 a^2= 11s-18

3. p^2=9p-14

Answer 1

1. b^2=10b-16

b^2 - 10b + 16 = 0

The coefficient of b (-10) is negative, and the constant (16) is positive. This means that the solution is of the form (b - x)(b - y), where x+y is 10 and xy is 16. So...

Think of the factors of 16 and come up with two numbers whose sum is 10. (4, 4) doesn't work. The answer is (8, 2), so the factored equation is:

(b - 8) (b - 2) = 0

Solutions are b=8 and b=2

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2. a^2 = 11a - 18

a^2 - 11a + 18 = 0

You have the same pattern with signs on the term with a and the constant term (negative, and positive, respectively).

So, this time you need two numbers whose sum is 11 and whose product is 18. (Considering the factors of 18: 9 and 2 works, 6 and 3 doesn't)

(a - 9) (a - 2) = 0

Solutions are a=9 and a=2

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3. p^2 = 9p - 14

p^2 - 9p + 14 = 0

Same pattern again. Need two numbers whose sum is 9 and whose product is 14. (7 and 2)

(p - 7)(p - 2) = 0

Solutions are p=7 and p=2.

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Note that the factoring process is slightly different when the sign on the coefficient of the x^1 term and the constant term is not the same as above. Also it is slightly different if there is a coefficient on the x^2 term.

Answer 2

You need to first get everything over on one side of the equation.

b^2 - 10b + 16

Now you can factor. You need to look for two numbers that when multiplied together will give you 16, but when added together will give you 10. The only numbers I can think of that would work for this would be 2 and 8. I also know that both the 2 and the 8 must be negative, since multiplying a negative 2 and a negative 8 would give me a positive 16 and adding a -2 and a -8 would give me -10. So, factored the answer would be:

(b - 8)(b -2)

So, b = 8 or b = 2

(This is because b - 8 = 0, so b - 8 + 8 = 0 + 8, b = 8)

Try the others and see if this helped, if not, get with me and let me know!

Answer 3

1) b^2=10b-16
b^2-10b+16=0
(b-2)(b-8)=0
b-2=0 b-8=0
b=2 b=8

2) (and i'm assuming that "s" is suppose to be an "a")
a^2= 11s-18
a^2-11s+18=0
(a-9)(a-2)=0
a=9 a=2

3) p^2=9p-14
p^2-9p+14=0
(p-7)(p-2)=0
p=7 p=2

Answer 4

1)

Rearrange to get;

b^2 - 10b -16 =0

b^2 - 2b - 8b -16 = 0

b (b - 2) - 8 (b - 2 ) = 0

(b - 8) (b - 2) = 0

b = 8 or 2


Questions is not possible as you have two unknowns.

Lets assume they are both a's

rearrange to get:

a^2 - 11a -18 = 0

a = 2 or 9

Last one

p = 2 or 7

The simple rule is get the factors of the number on it's own (They multiply to make it) and they must add together to make the other number attached to the letter.