SOLUTION: 1.x^4=5x^2 2.x^4+18x^3+8x+144=0 3.x^3+19x^2-7x-133=0 4.x^6-13x^5-7x=-91 I need help solving these. So far I have spent hours on just these problems our teacher doesn't explain

Question 156591: 1.x^4=5x^2
2.x^4+18x^3+8x+144=0
3.x^3+19x^2-7x-133=0
4.x^6-13x^5-7x=-91
I need help solving these. So far I have spent hours on just these problems our teacher doesn't explain things very well.
Thanks
Lauren
Found 2 solutions by stanbon, Earlsdon:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
1.x^4=5x^2
x^4 - 5x^2 =
x^2 (x^2 - 5) = 0
x^2 (x-sqrt(5))(x+sqrt(5)) = 0
x = 0 or x = sqrt(5) or x = -sqrt(5)
---------------------------------
2.x^4+18x^3+8x+144=0
x^3(x+18) + 8(x+18) = 0
(x+18)(x^3+8) = 0
(x+18)(x+2)(x^2-2x+4) = 0
x = -18 or x = -2 or x^2-2x+4 =0
---
Use the quadratic formula on x^2-2x+4
x = [2 +- sqrt(4 - 4*1*4)]/2
x = [2 +- sqrt(-12)]/2
x = [1 +- isqrt(3)]
----------------------------------
3.x^3+19x^2-7x-133=0
x^2(x + 19) -7(x+19) = 0
(x+19)(x^2-7) = 0
(x+19)(x-sqrt(7))(x+sqrt(7)) = 0
x = -19 or x=sqrt(7) or x=-sqrt(7)
----------------------------------
4.x^6-13x^5-7x=-91
x^6 - 13x^5 -7x + 91 = 0
x^5(x-13) - 7(x-13) = 0
(x-13)(x^5-7) = 0
x = 13 and x = 7^(1/5)
=============================
Cheers,
Stan H.

Answer by Earlsdon(6294) (Show Source): You can put this solution on YOUR website!
Solve for x:
1) Divide both sides by
Factor out an
Apply the zero product rule: If a*b = 0, then either a = 0 or b = 0 or both.
or
If then and
If then Take the square root of both sides to get:
or
Now because this is a fourth degree polynomial, you can expect four solutions, some of which may be identical.
The solutions are:

--------------------------------
2) This polynomial can be factored into:
and the first factor can be further factored as it is the sum of two cubes and , so we get:
Now, applying the zero product rule that was used in the first problem, you will set each of these three factors equal to zero.
Here,
Here, we'll have to use the quadratic formula to solve for x: and a = 1, b = -2, and c = 4, so...
Simplifying this, we get:

or Simplifying this:
or
...and for the last one:

So, as in the first problem, because you have a fourth degree polynomial, you can expect four solutions, and they are:

-----------------
3) You can solve this by "factoring by grouping"
Now factor each of the two terms:
Factor out the common factor of
Apply the zero product rule.
or so...
or
Take the square root of both sides.
or
The three solutions are:

----------------
I'll get you started on the last one but see if you can finish it up yourself.
4) Add 91 to both sides.
Factor by grouping.
Factor an from the first term and a 7 from the second term.
Now factor the common factor
Apply the zero product rule.
or
Can you finish this?
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