O-Level Additional Math

[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.

[ADoc]

suiyuan:

Hi


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

Please help.

Thank you.

Hi! I understand your concern regarding these two definitions.
Factorisation, I believe, is more intuitive in that it literally just means applying methods such as factorisation by groups to re-write the given expression into multiplication of two or more brackets (i.e. factors).

example, x²+x = (x)(x+1) is factorisation whereas
x²+x = (x+0.5)² - 0.5² isn't factorisation per se


Simplify usually goes with terms such as expand. It means to reduce an ugly looking expression into something simpler by first expanding, group the like terms, and finally add/subtract. All these to be done according to BODMAS or PEMDAS of course.

Some schools implicitly expects the students to factorise as factorised mathematical expressions appear simpler-looking.

To summarise:
(1) factorise is to form a strict multiplication of two or more brackets
(2) simplify is to reduce a long or ugly expression into something simpler or shorter (not always shorter sometimes), and we can leave the final answer in factorised form if we prefer [your teacher should not fault you if he/she is worth his/her two cents]

[ADoc]

suiyuan:

Sec Two Maths


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

(^2 means to the power of 2)

Please help.

Thank you.

Hi!
Just so you understand and subsequently apply the formula on your own, here's a long-winded explanation outlining the mental process you should have. So bear with me... :boogie:

(3x+7y)² – (-2x-3)² = (a + b)² - (c - d)² [you must be familiar with these formula already]

a = 3x
b = 7y
c = -2x
d = 3 (NOT negative 3 so that you won't confuse yourself unnecessarily)

(3x+7y)² – (-2x-3)² = (3x)² + 2(3x)(7y) + (7y)² - [ (-2x)² - 2(-2x)(3) + 3² ] [for inexperienced learners, I recommend inserting the square bracket to avoid committing the mistakes of -/+, else you can just switch the signs as you work] Always be extra careful whenever you see negative signs!!

I'll leave the rest to you.

[Herbie]

Hi! Are there rules in solving qn on factorisation?? Tq

[Herbie]

(3t+4s)(6q-7r)-(t-3s)(7r-6q)

Can some show the steps to such qn? Tq