O-Level Additional Math

[suiyuan]

Sec Two Maths


1) Given that x^2 = 5x – 1, find the value of 2x^4 + 2/x^4

2) a)Factorise 2x^10 – 13x^5 -15 completely.
b)Factorise 10x^10y – 65x^5y-75y

Note a) ^2 means to the power of 2 b) 2/x^4 means 2 divided by x^4

Please help.

Thank you

[CoffeeCat]

suiyuan:

Sec Two Maths


1) Given that x^2 = 5x – 1, find the value of 2x^4 + 2/x^4

2) a)Factorise 2x^10 – 13x^5 -15 completely.
b)Factorise 10x^10y – 65x^5y-75y

Note a) ^2 means to the power of 2 b) 2/x^4 means 2 divided by x^4

Please help.

Thank you

Do you happen to be from some IP school or sth?
Your teacher probably koped qns 1 from SMO.

Qns 1)
Lousy Method : solve for x, substitute and torture yourself in the process.
Proper Method : divide the quadratic equation by x throughout to get
x + 1/x = 5
You will get x^2 + 1/x^2 = 23 by squaring the above.
Repeat the process one more time.
Final ans should be 1054 if I have no calculation error.

Qns 2a)
Notice 10 is twice that of 5. hence imagine you substitute y=x^5. You will reduce the problem to factorising a quadratic expression.

Qns 2b)
Ambiguous.......Please use brackets.
10x^10y means 10x^(10y) or 10(x^10)y

Since it's part b, i will assume the latter.
First extract common factors from all terms, which is just 5y. After that what you have should be part a.

[JadeDry]

Hello again, I have a problem that I am unsure about:


Simplify the following expression:

3/4((x squared)(y)) X 5/9((x)(z cubed)) / ((20/9)(x quadrupled)(y squared)(z))

These are my workings:
3/4((x squared)(y)) X 5/9((x)(z cubed))=

15/36((x cubed)(y)(z cubed))


15/36((x cubed)(y)(z cubed)) / ((20/9)(x quadrupled)(y squared)(z))=

3(Z squared) / 16(xy)

Are my calculations correct?

On a side note, is Z cubed/ Z= Z squared or Z cubed?

Thanks in advance.

[suiyuan]

Sec Two Maths


Find the smallest possible value of the expression 3x^2 + 27y^2+5z^2-18xy-30z+125.

Note a) ^2 means to the power of 2

Please help.

Thank you

[FrekiWang]

suiyuan:

Sec Two Maths


Find the smallest possible value of the expression 3x^2 + 27y^2+5z^2-18xy-30z+125.

Note a) ^2 means to the power of 2

Please help.

Thank you

Note that
3x^2-18xy+27y^2=3(x^2-6xy+9y^2)=3(x-3y)^2
5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

So the original express is
(3x^2-18xy+27y^2)+(5z^2-30z)+125
=3(x-3y)^2+5(z-3)^2-45+125
=3(x-3y)^2+5(z-3)^2+80

Since the square of any number is non-negative (0 is the smallest possible), the minimum value of the whole expression is 80.

PS: A bit too much for a sec 2 student, probably from one of the top schools?

[suiyuan]

FrekiWang:

5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

Hi

Please explain in more details how you get this.

5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

Thank you

[CoffeeCat]

suiyuan:

FrekiWang:

5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

Hi

Please explain in more details how you get this.

5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

Thank you

He is basically completing the square. If your teacher posed that question, then he is assuming that you all know how to complete the square. Completing the square is a simple technique for solving simpler \"find minimum of value of a simple quadratic looking expression\".

You should be able to find completing the square in ur sec2 textbook.

[FrekiWang]

suiyuan:

FrekiWang:

5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

Hi

Please explain in more details how you get this.

5z^2-30z=5(z^2-6z)=5(z^2-6z+9-9)=5(z^2-6z+9)-45=5(z-3)^2-45

Thank you

I used to believe that 'completing the square' is in Exp Sec 3 syllabus.

Since you asked, here is the explanation.

You should have learnt that a^2-2ab+b^2=(a-b)^2 in Sec 2.

Compare this formula with z^2-6z. we assume z^2 is the a^2 term and 6z is the 2ab term, then b must be 3. Therefore, when we add a 3^2 (which is equal to 9) at the back, the expression z^2-6z+9 will become a complete square (z-3)^2. However, when we introduce an additional +9, we have to subtract a 9 to keep the whole expression unchanged.

[suiyuan]

Hi


Is there a difference between simplification and factorisation of algebraic expressions?

Please help.

Thank you.

[suiyuan]

Sec Two Maths


Expand and simplify
(3x+7y)^2 – (-2x-3)^2

(^2 means to the power of 2)

Please help.

Thank you.