I need algebra help!?

Question

I got a math packet for over the summer, and there are three questions that are giving me issues.

1) SOLVE EACH EQUATION. SHOW ALL WORK.

A. 1/2(6n+8) = 4n - (n-4)

B. 6(3-4t) = -12(2t-3/2)

2. SOLVE EACH EQUATION BY FACTORING. SHOW ALL WORK.

A. x^4 - 5x^2 + 4 = 0

I keep getting different answers each time. Can anyone help me?

Answer 1

A.  1/2(6n+8)=4n-(n-4)

    3n+4= 4n-n+4

     3n+4=3n+4

n=0

B. 6(3-4t)=-12(2t-3/2)

  18-24t=-24t-18

t= 0

x^4-5x^2+4=0

(x^2-1)(x^2-4)=0

(x=1)(x+1)(x-2)(x+2)=0

x=-1,1,-2, or 2

Answer 2

1.A. 1/2 times 6n + 8 = 3n + 4 You could do it by substituting .5 for 1/2 - .5 x 6n = 3.0n .5 x 8 = 4.0.  On the other side, all you have to do is sign work -(n-4) is the same as -1 times n and -1 times -4 which results in -n and +4 so you have 4n -n +4 which is 3n + 4.

1.B. You just need to multiply what's inside the parans by what's outside.  6 times 3 = 18 and 6 times -4t = -24t;  -12 times 2 t = -24t and -12 times -3/2 = 18.

2.  In order to factor an equation, you need to find terms that will satisfy all the parts of the equation.  Two factors that result in x^4 are x^2 times x^2.  Two factors that result in +4 are 2 times 2 and 1 times 4 or -2 times -2 and -1 times -4.  Only one set of factors add up to -5. those are -1 and -4.  Therefore, the equation factors into (x^2 -1) (x^2 -4)

Answer 3

on the first problem you can distribute the 1/2 or multiply both sides of the equation by the reciprocal of 1/2.  I will distribute first;

3n - 4 = 4n  - n + 4

3n - 4 = 3n - 4

this ism true for all values of n.

now muiltiply both sides by the reciprocal.

6n - 8 =  2 [ 4n - ( n - 4 ) ]

6n - 8 = 2[ 4n - n - 4 ]

6n - 8 = 2 [ 3n - 4 ]

6n - 8 = 6n - 8

this is true for all values of n

b) 18 - 24t = -24t + 18

   this is the same situation as the first problem.  You would end up with 0 = 0 , therefore the equation is true for all values of t.

x ^4 - 5x^2 + 4 = 0

( x² - 4) ( x² - 1 ) = 0

( x + 2 )( x - 2 ) ( x + 1 ) ( x - 1 ) = 0

x = ± 2 or x = ± 1

Answer 4

1) a

1/2(6n+8) = 4n - (n-4)

3n+4=4n-n+4

3n+4=3n+4

There is no answer because the two sides of the equation are the same, n could be any number

b) 6(3-4t)= -12(2t-3/2)

3-4t = -2(2t-3/2)

3-4t = -4t + 3

3-4t = 3-4t, same ans in a) t could be anything and it will work

2) x^4 - 5x^2 +4 = 0

so, think of a factor and look for factors in the terms

The numbers to look at are 1 for the first term, 5 for the second and 4 for the last.  Factors of 1 are 1, factors of 5 could be 1 and 4 or 2 and 3, factors of 4 are 1 and 4 or 2 and 2.  Look at the terms and make a good guess:

(x^2- 1)(x^2-4)=0

Backchecking we can muliply this out and make sure this is correct

x^4 -4 x^2 -x^2 +4 = 0

X^4 -5x^2 +4 =0

Which is right

Takes some fiddling to figure out the factors, its not a "plug and chug" kind of thing, try and check, try and check, you'll solve it!

Ok now keep going to make it even simpler

(x^2-1)(x^2-4) = 0

Factors into

(x+1)(x-1)(x+2)(x-2) = 0

Now look for things that would make the whole term =0 and since you are multiplying you just need 1 term of the 4 to be zero make it all zero

So if x = -1 (first term) or x = 1 (second term) or x=-2 (third term) or x = 2 (fourth term, one term would be zero and zero the whole equation.   So the answer is that x = (-1, 1, -2, 2)

You can feed these answers back into the original equation and get

(-1)^4 - 5(-1)^2 +4 =0

1 -5 +4 = 0

0=0, true

Just do the same with the other 3 to be sure and yer done!

Answer 5

A.

1/2(6n+8) = 4n - (n-4)

(3n+4) = 4n - n + 4

3n - 4n + n = 4 - 4

0n = 0

n can not be figured out?

B.

6(3-4t) = -12(2t-3/2)

18 - 24t = -24t + 18

18 -18  = -24t + 24t

0 = 0t

t can not be figured out?

2a.

x^4 - 5x^2 + 4 = 0

(x^2  -1)(x^2  -4)

either:

x^2  -1 = 0

x^2  = 1

x = +/- square root of 1

or:

x^2  -4 = 0

x^2 = 4

x = +/- 2

Answer 6

1) SOLVE EACH EQUATION. SHOW ALL WORK.

A. 1/2(6n+8) = 4n - (n-4)

6/2n + 8/2 = 4n - n + 4

3n + 4 = 3n +4

B. 6(3-4t) = -12(2t-3/2)

18 - 24t = -24t + 36/2

-24t + 18 = -24t + 18

2. SOLVE EACH EQUATION BY FACTORING. SHOW ALL WORK.

A. x^4 - 5x^2 + 4 = 0

(x^2 - 4)(x^2 - 1) = 0

(x - 2)(x + 2)(x - 1)(x + 1) = 0

(x - 2) = 0

x = 2

(x + 2) = 0

x = -2

(x - 1) = 0

x = 1

(x + 1) = 0

x = -1

Answer 7

1)1/2(6n+8)=4n-n+4   1/2(6n+8)=3n+4  cross multiply

   6n+8=2(3n+4)  6n+8=6n+8   6n-6n=8-8  0=0 =0 (funny answer!)

1b)18-24t   =-24t+(12*3)/2 = 18-24t=-24t+18  =0(funny answer)

note 1a an 1b are not right as n  or t is not equal to 0

Answer 8

1.

3n+4=3n+4

n=0

Answer 9

1A.

1/2 (6n + 8) = 4n -- (n -- 4)

3n + 4 = 4n -- n + 4

3n + 4 = 3n + 4

The given equation is an identity (that is, always true, no matter what value n takes). The equation has indefinitely many solutions.

1B.

6 (3 -- 4t) = --12 (2t -- 3/2)

18 -- 24t = --24t + 18

The equation has indefinitely many solutions.

2.

x^4 -- 5x^2 + 4 = 0

Put u = x^2 to bring the 4th-degree polynomial equation to a quadratic equation:

u^2 --5u + 4 = 0

Solve for u:

u1 = [5 -- sqrt(25 --16)]/2 = 1

u2 = [5 + sqrt(25 -- 16)]/2 = 4

Back to x:

Since u = x^2 ==> x = ±√u

The given equation has 4 solutions:--2, --1, +1, +2