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How To Calculate Quickly And Correctly In Mathematics - Education (2) - Nairaland 44e1v

How To Calculate Quickly And Correctly In Mathematics (11085 Views)

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Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 7:47pm On Jun 18, 2013
Ortarico:

Yes you're right, cause with the options it's easier to work out what/how the examiner wants the equation to be.

A little error there Chief Ortarico

=> log25x5 - log1/255
=> log(5)^2(x)5 - log(25)(-1)5
=> (2x)log55 - log(5)2(-1)5
=> (2x)log55 - (-2)log(5)5
Since log55 = 1
=> (2x)log55 - (-2)log(5)5
=> 2x .(1) + 2
=> 2x + 2
=> 2(x + 1)

So the final answer should be => 2(x + 1) which is option B as solved by Omosivie

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Ortarico(m): 7:51pm On Jun 18, 2013
doubleDx:

A little error there Chief Ortarico

=> log25x5 - log1/255
=> log(5)^2(x)5 - log(25)(-1)5
=> (2x)log55 - log(5)2(-1)5
=> (2x)log55 - (-2)log(5)5
Since log55 = 1
=> (2x)log55 - (-2)log(5)5
=> 2x .(1) + 2
=> 2x + 2
=> 2(x + 1)

So the final answer should => 2(x + 1)

Ok thanks general doubleDx

1 Like

Calculusfx: 8:52am On Jun 19, 2013
To find the sum of positive integers...1+2+3+...+n=n(n+1)/2...
Omosivie1(f): 2:32pm On Jun 19, 2013
Pls help me solve this: 1/root 2 -1 minus 1/root 2 +1. Thanks
Ortarico(m): 4:56pm On Jun 19, 2013
Omosivie,:
Pls help me solve this: 1/root 2 -1 minus 1/root 2 +1. Thanks

NB: _/ is for the root
1/(_/2 - 1) - 1/(_/2 + 1)
The LCM is (_/2 - 1)(_/2 + 1):
=> (_/2 + 1) - (_/2 - 1)/(_/2 - 1)(_/2 + 1)
Clearing the bracket we'll get:
=> _/2 + 1 - _/2 + 1/_/4 + _/2 - _/2 - 1

=> 1 + 1/2 - 1
=> 2. . . . . . . .ans

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Omosivie1(f): 6:36pm On Jun 19, 2013
Ortarico:

NB: _/ is for the root
1/(_/2 - 1) - 1/(_/2 + 1)
The LCM is (_/2 - 1)(_/2 + 1):
=> (_/2 + 1) - (_/2 - 1)/(_/2 - 1)(_/2 + 1)
Clearing the bracket we'll get:
=> _/2 + 1 - _/2 + 1/_/4 + _/2 - _/2 - 1

=> 1 + 1/2 - 1
=> 2. . . . . . . .ans
Omosivie1(f): 1:28pm On Jun 21, 2013
Pls help me out with this two simultaneous equations:
1, xy=-3; x+y=2
2, x+2y=0; x^2+y^2=20
Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 6:54pm On Jun 21, 2013
Omosivie,:
Pls help me out with this two simultaneous equations:
1, xy=-3; x+y=2
2, x+2y=0; x^2+y^2=20
wat r d opti0ns?
Ortarico(m): 8:06pm On Jun 21, 2013
Omosivie,:
Pls help me out with this two simultaneous equations:
1, xy=-3; x+y=2
2, x+2y=0; x^2+y^2=20

Let's go. . . .
1. xy = -3. . . . . .eqn(i)
x + y= 2. . . . . . .eqn(ii)
from eqn(ii):
x= 2 - y. . . . . .eqn(iii)
substitute eqn(iii) in2 eqn(i):
xy= -3
(2 - y) y= -3
2y - y2= -3
Re-arrangng gives:
y2 - 2y - 3= 0
(y - 3)(y + 1)
y= 3 or -1
:. x= 1 or -3
To check:
xy= -3; (-1*3)= -3
x + y= 2; (-1 + 3)= 2

2. x + 2y= 0. . . . . .eqn(i)
x2 + y2= 20. . . . .eqn(ii)
from eqn(i):
x + 2y= 0
x= -2y. . . . . . .eqn(iii)
substitute eqn(iii) in2 eqn(ii):
x2 + y2= 20
(-2y)2 + y2= 20
4y2 + y2= 20
5y2= 20
y2= 4
y= +/- 2
substitute y= 2 in2 eqn(i):
x + 2y= 0
x + 2(2)= 0
x + 4= 0
x= -4
To check:
x + 2y= 0; -4 + 2(2)= 0
x2 + y2= 20; (-4)2 + (2)2= 20

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Omosivie1(f): 8:42pm On Jun 21, 2013
Ortarico:

Let's go. . . .
1. xy = -3. . . . . .eqn(i)
x + y= 2. . . . . . .eqn(ii)
from eqn(ii):
x= 2 - y. . . . . .eqn(iii)
substitute eqn(iii) in2 eqn(i):
xy= -3
(2 - y) y= -3
2y - y2= -3
Re-arrangng gives:
y2 - 2y - 3= 0
(y - 3)(y + 1)
y= 3 or -1
:. x= -1 or 3
To check:
xy= -3; (-1*3)= -3
x + y= 2; (-1 + 3)= 2

2. x + 2y= 0. . . . . .eqn(i)
x2 + y2= 20. . . . .eqn(ii)
from eqn(i):
x + 2y= 0
x= -2y. . . . . . .eqn(iii)
substitute eqn(iii) in2 eqn(ii):
x2 + y2= 20
(-2y)2 + y2= 20
4y2 + y2= 20
5y2= 20
y2= 4
y= 2
substitute y= 2 in2 eqn(i):
x + 2y= 0
x + 2(2)= 0
x + 4= 0
x= -4
To check:
x + 2y= 0; -4 + 2(2)= 0
x2 + y2= 20; (-4)2 + (2)2= 20
Thanks, but the answer is not the same with my textbook's answer. According to my textbook, the answer for number one is -3 or 1 and the answer for number two is (4,2) or (-4,2). I don't understand dis textbook and i'm totally fed up.
Omosivie1(f): 8:45pm On Jun 21, 2013
Leebliss13: wat r d opti0ns?
No options
Ortarico(m): 8:50pm On Jun 21, 2013
Omosivie,:
Thanks, but the answer is not the same with my textbook's answer. According to my textbook, the answer for number one is -3 or 1 and the answer for number two is (4,2) or (-4,2). I don't understand dis textbook and i'm totally fed up.

But that's it now.
At no 1 x is (-3 , 1) then y is (-1 , 3) and at no 2 x is -4, then y is 2. To check that you saw was just to confirm. That's what I got sister

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Omosivie1(f): 9:17pm On Jun 21, 2013
Ortarico:

But that's it now.
At no 1 x is (-3 , 1) then y is (-1 , 3) and at no 2 x is -4, then y is 2. To check that you saw was just to confirm. That's what I got sister
Thanks, I definately need glasses
Ortarico(m): 9:42pm On Jun 21, 2013
Omosivie,:
Thanks, I definately need glasses

Thanks too, it happens to everyone.
Calculusfx: 4:25pm On Jun 22, 2013
this bro,the square root of a natural number is + or -...so,from where you got y^2=4...therefore y=+ or - 2...therefore y=+2 or -2...from the first equation x+2y=0...substitute y=-2...therefore x+2(-2)=0...x-4=0...x=4...substitute y=2...x+2y=0...x+2(2)=0...x+4=0...x=-4...therefore,when y=-2,x=4 and when y=2,x=-4
Ortarico(m): 4:51pm On Jun 22, 2013
Calculusf(x):
this bro,the square root of a natural number is + or -...so,from where you got y^2=4...therefore y=+ or - 2...therefore y=+2 or -2...from the first equation x+2y=0...substitute y=-2...therefore x+2(-2)=0...x-4=0...x=4...substitute y=2...x+2y=0...x+2(2)=0...x+4=0...x=-4...therefore,when y=-2,x=4 and when y=2,x=-4

All right, but if you look at it well, you'll see that I typed +/- 2, I know the rule boss.

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Omosivie1(f): 1:46pm On Jun 24, 2013
I hope i'm not bothering u guys with these "easy" questions but can u please help me out with them
1, m^3 + mn^2-nm^2-n^3/m^4-n^4
2, -12a^2x/-8a^2b^2
Ortarico(m): 2:39pm On Jun 24, 2013
Omosivie,:
I hope i'm not bothering u guys with these "easy" questions but can u please help me out with them
1, m^3 + mn^2-nm^2-n^3/m^4-n^4
2, -12a^2x/-8a^2b^2

1. m3 + mn2 - nm2 - n3/m4 - n4
m3 + n3 + mn2 - nm2/m4 - n4
(m - n)3 + mn(n - m)/(m - n)4
mn(n - m)/(m - n) . . . . .ans
Ortarico(m): 2:40pm On Jun 24, 2013
2. Please re-type, there's a mistake or I don't get you well.
Omosivie1(f): 3:09pm On Jun 24, 2013
Ortarico: 2. Please re-type, there's a mistake or I don't get you well.
-12a^2 x / -8a^2 b^2
Omosivie1(f): 3:17pm On Jun 24, 2013
Ortarico:

1. m3 + mn2 - nm2 - n3/m4 - n4
m3 + n3 + mn2 - nm2/m4 - n4
(m - n)3 + mn(n - m)/(m - n)4
mn(n - m)/(m - n) . . . . .ans

The answer @ the back of my textbook is 1/m+n
Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 6:47pm On Jun 24, 2013
Omosivie,:
I hope i'm not bothering u guys with these "easy" questions but can u please help me out with them
1, m^3 + mn^2-nm^2-n^3/m^4-n^4


{m3 + mn2 - nm2 - n3}/(m4 - n4)

Simplifying m4 - n4 yields => (m2^2 - n2^2 => (m2 - n2)(m2 + n2)...difference of 2 squares.

Rearranging the numerator & factorising yields =>
=> mn2 - nm2 + m3 - n3
=> mn(n - m) - (n3 - m3)
Now, (n3 - m3) can be rewritten as => (n - m)3 + 3mn(n - m) - difference of two cubes, which makes the numerator => mn(n - m) - {(n - m)3 + 3mn(n - m)}

Putting the numerator & denominator together yields:

=>> [ mn((n - m) - {(n - m})3 + 3mn((n - m))]/(m2 - n2)(m2 + n2)

Factorising (n - m) from the numerator above yields=>

=>> [(n - m)][mn - {(n - m)2 + 3mn}]/(m2 - n2)(m2 + n2)

=>> [n - m][mn - {(n2 - 2mn + m2 + 3mn)]/(m2 - n2)(m2 + n2)

=>> [n - m][mn - {(n2 + m2 + mn)]/(m2 - n2)(m2 + n2)

=>> (n - m)(-n2 - m2)/(m2 - n2)(m2 + n2)

=>> (n - m)(-1)(n2 + m2)/(m2 - n2)(m2 + n2)
=>> (m - n)(n2 + m2)/(m2 - n2)(m2 + n2)


=>> (m - n)/(m2 - n2)

=>> (m - n)/(m - n)(m + n)

=>> 1/(m + n)

Hence,

{m3 + mn2 - nm2 - n3}/(m4 - n4) = 1/(m + n)

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Omosivie1(f): 7:12pm On Jun 24, 2013
doubleDx:

{m3 + mn2 - nm2 - n3}/(m4 - n4)

Simplifying m4 - n4 yields => (m2^2 - n2^2 => (m2 - n2)(m2 + n2)...difference of 2 squares.

Rearranging the numerator & factorising yields =>
=> mn2 - nm2 + m3 - n3
=> mn(n - m) - (n3 - m3)
Now, (n3 - m3) can be rewritten as => (n - m)3 + 3mn(n - m) - difference of two cubes, which makes the numerator => mn(n - m) - {(n - m)3 + 3mn(n - m)}

Putting the numerator & denominator together yields:

=>> [mn[b](n - m)[/b] - {(n - m})3 + 3mn((n - m))]/(m2 - n2)(m2 + n2)

Factorising (n - m) from the above yields=>

=>> [(n - m)][mn - {(n - m)2 + 3mn}]/(m2 - n2)(m2 + n2)

=>> [n - m][mn - {(n2 - 2mn + m2 + 3mn)]/(m2 - n2)(m2 + n2)

=>> [n - m][mn - {(n2 + m2 + mn)]/(m2 - n2)(m2 + n2)

=>> (n - m)(-n2 - m2)/(m2 - n2)(m2 + n2)

=>> (n - m)(-1)(n2 + m2)/(m2 - n2)(m2 + n2)
=>> (m - n)(n2 + m2)/(m2 - n2)(m2 + n2)


=>> (m - n)/(m2 - n2)

=>> (m - n)/(m - n)(m + n)

=>> 1/(m + n)

Hence,

{m3 + mn2 - nm2 - n3}/(m^4 - n^4) = 1/(m + n)
Wow!!!! This is so complex(one of the reasons why I hate maths). Could this be further maths?? I just pray I'm not given this question in my exam cos it's too long, difficult and complex. Thanks for solving it.
Omosivie1(f): 7:14pm On Jun 24, 2013
@ Ortarico, Calucus(fx) and DoubleDx. Pls help me out with the number 2 question
Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 7:33pm On Jun 24, 2013
Omosivie,:
Wow!!!! This is so complex(one of the reasons why I hate maths). Could this be further maths?? I just pray I'm not given this question in my exam cos it's too long, difficult and complex. Thanks for solving it.

It's not Further Maths lol, the solution is not as long as it appears, though my explanation is, so you could understand better. It's just a matter of understanding two expressions and using them at the right time =>

1. That A2 - B2 = (A - B)(A + B) - difference of two squares
2. That A3 - B3 = (A - B)3 - 3AB(A - B) - difference of two cubes

So that A4 - B4 = (P2 - Q2) => (P - Q)(P + Q), such that =>> P = A2 and Q = B2.

Understanding these expressions and knowing when to apply/fix them in when you factorize makes the question weak and easy to solve.

I hope you catch my drift... so, you see? It's easy, don't be discouraged god sister! grin

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Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 7:47pm On Jun 24, 2013
Omosivie,:
-12a^2 x / -8a^2 b^2

Solution

-12a2 x /-8a2 b2

=> This is easy seeing as -4a2 is a factor of the numerator and denomitor, cancelling out & reducing the expression to =>

For the numerator => -12a2/-4a2 = 3x
For the denomitor => -8a2 b2/-4a2 = 2b2

Putting the simplified numerator and denomitor together yields =>

=> 3x/2b2
Ortarico(m): 8:01pm On Jun 24, 2013
doubleDx:

{m3 + mn2 - nm2 - n3}/(m4 - n4)

Simplifying m4 - n4 yields => (m2^2 - n2^2 => (m2 - n2)(m2 + n2)...difference of 2 squares.

Rearranging the numerator & factorising yields =>
=> mn2 - nm2 + m3 - n3
=> mn(n - m) - (n3 - m3)
Now, (n3 - m3) can be rewritten as => (n - m)3 + 3mn(n - m) - difference of two cubes, which makes the numerator => mn(n - m) - {(n - m)3 + 3mn(n - m)}

Putting the numerator & denominator together yields:

=>> [ mn((n - m) - {(n - m})3 + 3mn((n - m))]/(m2 - n2)(m2 + n2)

***** Factorising (n - m) from the numerator above yields=>

=>> [(n - m)][mn - {(n - m)2 + 3mn}]/(m2 - n2)(m2 + n2)

=>> [n - m][mn - {(n2 - 2mn + m2 + 3mn)]/(m2 - n2)(m2 + n2)

=>> [n - m][mn - {(n2 + m2 + mn)]/(m2 - n2)(m2 + n2)

=>> (n - m)(-n2 - m2)/(m2 - n2)(m2 + n2)

=>> (n - m)(-1)(n2 + m2)/(m2 - n2)(m2 + n2)
=>> (m - n)(n2 + m2)/(m2 - n2)(m2 + n2) *****


=>> (m - n)/(m2 - n2)

=>> (m - n)/(m - n)(m + n)

=>> 1/(m + n)

Hence,

{m3 + mn2 - nm2 - n3}/(m4 - n4) = 1/(m + n)

Respect to you @ general doubleDx. I supposed to have used the law of cubic and quadratic expression but for my hurry I missed it.
Please help explain more at the five starred phase of your work, thanks general!

1 Like

Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 9:12pm On Jun 24, 2013
Ortarico:

Respect to you @ general doubleDx. I supposed to have used the law of cubic and quadratic expression but for my hurry I missed it.
Please help explain more at the five starred phase of your work, thanks general!

[ mn (n - m) - {(n - m)3 + 3mn (n - m)}]/(m2 - n2)(m2 + n2)

If you look at the numerator of the above expression carefully, you will realize that (n - m) is a common factor/coefficient @ mn (n - m) , (n - m)3 and 3mn (n - m)

So that if (n - m) is replaced with say Q, the numerator would be => [ mn Q - {(Q)3 + 3mn(Q)}] which reduces the numerator to =>

mnQ - {Q3 + 3mnQ}
mnQ - Q3 - 3mnQ

Now, if Q is factorized from the above, it becomes

=> Q(mn - Q2 - 3mn)
=> Q(-2mn - Q2)
=> -Q(2mn + Q2)

that Q = (n - m), so that -Q = (m - n), putting back the values of Q & -Q into the numerator's expression, yields =>

(m - n)(2mn + (n - m)2)
(m - n)(2mn + n2+ m2 - 2mn)
(m - n)(n2 + m2)

Putting the simplified numerator back into the expression yields =>

=>> (m - n)(n2 + m2)/(m2 - n2)(m2 + n2)

=>> (m - n)/(m2 - n2)

=>> (m - n)/(m - n)(m + n)

=>> 1/(m + n)

I hope this is clearer...

2 Likes

Ortarico(m): 9:41pm On Jun 24, 2013
doubleDx:

[ mn (n - m) - {(n - m)3 + 3mn (n - m)}]/(m2 - n2)(m2 + n2)

If you look at the numerator of the above expression carefully, you will realize that (n - m) is a common factor/coefficient @ mn (n - m) , (n - m)3 and 3mn (n - m)

So that if (n - m) is replaced with say Q, the numerator would be => [ mn Q - {(Q)3 + 3mn(Q)}] which reduces the numerator to =>

mnQ - {Q3 + 3mnQ}
mnQ - Q3 - 3mnQ

Now, if Q is factorized from the above, it becomes

=> Q(mn - Q2 - 3mn)
=> Q(-2mn - Q2)
=> -Q(2mn + Q2)

that Q = (n - m), so that -Q = (m - n), putting back the values of Q & -Q into the numerator's expression, yields =>

(m - n)(2mn + (n - m)2)
(m - n)(2mn + n2+ m2 - 2mn)
(m - n)(n2 + m2)

Putting the simplified numerator back into the expression yields =>

=>> (m - n)(n2 + m2)/(m2 - n2)(m2 + n2)

=>> (m - n)/(m2 - n2)

=>> (m - n)/(m - n)(m + n)

=>> 1/(m + n)

I hope this is clearer...

Oh yes, thanks boss, one love!

1 Like

Calculusfx: 9:45pm On Jun 24, 2013
You have really done a great job@general doubledx...
Re: How To Calculate Quickly And Correctly In Mathematics by Nobody: 10:56pm On Jun 24, 2013
^

Thanks bruv!
Omosivie1(f): 9:31am On Jun 25, 2013
doubleDx:

It's not Further Maths lol, the solution is not as long as it appears, though my explanation is, so you could understand better. It's just a matter of understanding two expressions and using them at the right time =>

1. That A2 - B2 = (A - B)(A + B) - difference of two squares
2. That A3 - B3 = (A - B)3 - 3AB(A - B) - difference of two cubes

So that A4 - B4 = (P2 - Q2) => (P - Q)(P + Q), such that =>> P = A2 and Q = B2.

Understanding these expressions and knowing when to apply/fix them in when you factorize makes the question weak and easy to solve.

I hope you catch my drift... so, you see? It's easy, don't be discouraged god sister! grin
Thanks, I'll cram these algebraic identities/expressions.

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