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Answer to x^4-10x^2+9=0
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Kaeten
Hybrid Syndicate
#1 - 2011-09-15 10:21:42 UTC
And explanation please!

Forgotten how to do this crap after 4 years.
Alpheias
Tactical Farmers.
Pandemic Horde
#2 - 2011-09-15 10:29:35 UTC
You are in school to learn.

Agent of Chaos, Sower of Discord.

Don't talk to me unless you are IQ verified and certified with three references from non-family members. Please have your certificate of authenticity on hand.
Ghaztehschmexeh
Caldari Provisions
Caldari State
#3 - 2011-09-15 10:31:58 UTC
BODMAS

Brackets

Orders (ie Powers and Square Roots, etc.)

Division and Multiplication (left-to-right)

Addition and Subtraction (left-to-right)
Kaeten
Hybrid Syndicate
#4 - 2011-09-15 10:32:39 UTC
Alpheias wrote:
You are in school to learn.

My pal needs it, I'm a programmer with a job mate.Blink
Alpheias
Tactical Farmers.
Pandemic Horde
#5 - 2011-09-15 10:54:23 UTC
Kaeten wrote:
Alpheias wrote:
You are in school to learn.

My pal needs it, I'm a programmer with a job mate.Blink


And I am the Queen of England.

Agent of Chaos, Sower of Discord.

Don't talk to me unless you are IQ verified and certified with three references from non-family members. Please have your certificate of authenticity on hand.
Kaeten
Hybrid Syndicate
#6 - 2011-09-15 11:23:52 UTC
Alpheias wrote:
Kaeten wrote:
Alpheias wrote:
You are in school to learn.

My pal needs it, I'm a programmer with a job mate.Blink


And I am the Queen of England.


It's a lady that received a book on Math Course C (matte C in Swedish), is this horrible edition they apparently forgot to include an example of a 4th dimension equation (do not ask me), she is asking for the explanation, the answer is printed in black and white..

-1
1
-3
3
Louis deGuerre
The Dark Tribe
#7 - 2011-09-15 13:18:31 UTC
My God, the answer is so blatantly obvious.

CANT YOU SEE ??? ARE YOU BLIND ???

I'll give a hint

its a binary digit, fool.

Kids these days...
Taedrin
Federal Navy Academy
Gallente Federation
#8 - 2011-09-15 13:42:36 UTC  |  Edited by: Taedrin
Presuming you are in school, you need to know the process, not just the answer.

You have x^4 - 10x^2 + 9 = 0

There are two ways to solve this problem: 1) Factoring 2) Quadratic Equation + Substitution

FACTORING:
We will presume we can factor to:
(x^2 + a)(x^2 + b) = 0

We want to find 'a' and 'b' such that a*b = 9
and such that a + b = -10

Presuming that a and b are both integers, we have the following possibilities to satisfy a*b = 9
1, 9 : a + b = 10
-1,-9 : a + b = -10
3,3 : a + b = 6
-3,-3 : a + b = -6

You will notice that for a = -1 and b = -9 (or vice versa, order doesn't matter), a + b = -10 is satisfied.

We therefore have (x^2 - 1)(x^2 - 9) = 0
The roots for this equation are: x^2 = 1 and x^2 = 3

We take the square root of both sides, and we get:
x = ±1 and x = ±3
Note that we have 4 possible solutions in our solution set. This is because our polynomial is of degree 4. The Fundamental Theorem of Algebra allows us to conclude that every polynomial of degree n has exactly n (possibly complex) roots.

SUBSTITUTION + QUADRATIC FORMULA:
We have x^4 - 10x^2 + 9 = 0
Substitute t = x^2
t^2 - 10t + 9 = 0

Use:
a = 1, b = -10, c = 9
(+10 ± ((-10)^2 - 4*1*9)^(1/2)) / (2*1)

(10 ± (100 - 36)^(1/2)) / 2
(10 ± 8) / 2
18 / 2 or 2 / 2
t = 9 or t = 1

Since we substituted t = x^2, we have:
x^2 = 9 or x^2 = 1

The rest is the same as the factoring method above.

NOTE:
The factoring method that I presented made some critical assumptions, which are not always true: 1) that the roots of the polynomial were all integers 2) That you could factor the polynomial into a product of two binomials. Things get much more complicated if these two assumptions are not true. It all depends upon how cruel your textbook is.
Kaeten
Hybrid Syndicate
#9 - 2011-09-15 19:16:15 UTC  |  Edited by: Kaeten
Taedrin wrote:
Presuming you are in school, you need to know the process, not just the answer.

You have x^4 - 10x^2 + 9 = 0

There are two ways to solve this problem: 1) Factoring 2) Quadratic Equation + Substitution

FACTORING:
We will presume we can factor to:
(x^2 + a)(x^2 + b) = 0

We want to find 'a' and 'b' such that a*b = 9
and such that a + b = -10

Presuming that a and b are both integers, we have the following possibilities to satisfy a*b = 9
1, 9 : a + b = 10
-1,-9 : a + b = -10
3,3 : a + b = 6
-3,-3 : a + b = -6

You will notice that for a = -1 and b = -9 (or vice versa, order doesn't matter), a + b = -10 is satisfied.

We therefore have (x^2 - 1)(x^2 - 9) = 0
The roots for this equation are: x^2 = 1 and x^2 = 3

We take the square root of both sides, and we get:
x = ±1 and x = ±3
Note that we have 4 possible solutions in our solution set. This is because our polynomial is of degree 4. The Fundamental Theorem of Algebra allows us to conclude that every polynomial of degree n has exactly n (possibly complex) roots.

SUBSTITUTION + QUADRATIC FORMULA:
We have x^4 - 10x^2 + 9 = 0
Substitute t = x^2
t^2 - 10t + 9 = 0

Use:
a = 1, b = -10, c = 9
(+10 ± ((-10)^2 - 4*1*9)^(1/2)) / (2*1)

(10 ± (100 - 36)^(1/2)) / 2
(10 ± 8) / 2
18 / 2 or 2 / 2
t = 9 or t = 1

Since we substituted t = x^2, we have:
x^2 = 9 or x^2 = 1

The rest is the same as the factoring method above.

NOTE:
The factoring method that I presented made some critical assumptions, which are not always true: 1) that the roots of the polynomial were all integers 2) That you could factor the polynomial into a product of two binomials. Things get much more complicated if these two assumptions are not true. It all depends upon how cruel your textbook is.
<3

Thank you very much, this is exactly what she needed! My pal thanks you alot, and as you said exactly, for one to learn one must understand the mechanics. The other ppl in this thread expect my pal to for an example; look at iron and make a car out of it without knowing anything about molding iron, and the building the damn thing.

To the rest of the retards who think I'm a kid, I'm a programmer at my own webdesign shop http://www.thechestdesign.com currently programming for Rasta Star Energy Drink company at http://www.rastastar.se and is just about to release my own developed webshop+CMS to them.

Love 'adults' who are about as bright as 12 year olds.
Louis deGuerre
The Dark Tribe
#10 - 2011-09-15 22:33:29 UTC
Kaeten wrote:
Love 'adults' who are about as bright as 12 year olds.


...and still brighter than you Twisted
Headerman
Native Freshfood
Minmatar Republic
#11 - 2011-09-16 06:40:32 UTC
Simplifying
x4 + -10x2 + 9 = 0

Reorder the terms:
9 + -10x2 + x4 = 0

Solving
9 + -10x2 + x4 = 0

Solving for variable 'x'.

Factor a trinomial.
(1 + -1x2)(9 + -1x2) = 0

Factor a difference between two squares.
((1 + x)(1 + -1x))(9 + -1x2) = 0

Factor a difference between two squares.
((3 + x)(3 + -1x))(1 + x)(1 + -1x) = 0

Subproblem 1
Set the factor '(1 + x)' equal to zero and attempt to solve:

Simplifying
1 + x = 0

Solving
1 + x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + x = 0 + -1

Combine like terms: 1 + -1 = 0
0 + x = 0 + -1
x = 0 + -1

Combine like terms: 0 + -1 = -1
x = -1

Simplifying
x = -1
Subproblem 2
Set the factor '(1 + -1x)' equal to zero and attempt to solve:

Simplifying
1 + -1x = 0

Solving
1 + -1x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + -1x = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -1x = 0 + -1
-1x = 0 + -1

Combine like terms: 0 + -1 = -1
-1x = -1

Divide each side by '-1'.
x = 1

Simplifying
x = 1
Subproblem 3
Set the factor '(3 + x)' equal to zero and attempt to solve:

Simplifying
3 + x = 0

Solving
3 + x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + x = 0 + -3
x = 0 + -3

Combine like terms: 0 + -3 = -3
x = -3

Simplifying
x = -3
Subproblem 4
Set the factor '(3 + -1x)' equal to zero and attempt to solve:

Simplifying
3 + -1x = 0

Solving
3 + -1x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + -1x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1x = 0 + -3
-1x = 0 + -3

Combine like terms: 0 + -3 = -3
-1x = -3

Divide each side by '-1'.
x = 3

Simplifying
x = 3
Solution
x = {-1, 1, -3, 3}

Australian Fanfest Event https://forums.eveonline.com/default.aspx?g=posts&find=unread&t=90062
Sidus Isaacs
Center for Advanced Studies
Gallente Federation
#12 - 2011-09-16 10:32:42 UTC  |  Edited by: Sidus Isaacs
Never heard of wolfram alpha?

http://www.wolframalpha.com/input/?i=x^4-10x^2%2B9%3D0

(no proper link because the forum could not parse it...)
Taedrin
Federal Navy Academy
Gallente Federation
#13 - 2011-09-16 15:15:12 UTC
Headerman wrote:


Factor a trinomial.
(1 + -1x2)(9 + -1x2) = 0



The problem with your proposed solution is here. Factoring is a hard problem which many students have difficulty with. Simply saying "factor" and giving the factors is very harmful to students. Students need to know what clues you saw which hinted at the factors you gave.

This is similar to how students have difficulty doing integrals in a second semester Calculus class - people simply give them answers or maybe even tell them "do a trig substitution here" - but very few people actually tell students what clues you saw which caused you to even think about a trig substitution.