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Quadratic Equations – So Easy, My Grandma Can’t Do It

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Tuition given in the topic of A-Maths Tuition Questions from the desk of Miss Loi at 4:43 pm (Singapore time)

Updated on

To the Mathematics aristocrats who feel that questions here are getting “so easy, my grandma can do it!“: It’s now crunchtime for the O-Levels, not the Maths Olympiad. Time to forget that killer prelim paper from your elite school. Instead, it’s time to focus on getting that distinction by simply answering more standard TYS questions correctly than 75% of your cohort.

Though modern science has shown that Singaporean math students are the best in the world, the O-Levels are really just a collection of standard questions for the vast global student population. And to many, these seemingly easy questions could be cases of “so difficult, my grandma will kill me!“.

Instead, more pressing are the massive, massive numbers of careless mistakes tragically commited by students each year. That’s where Miss Loi comes in now – to nag you to death forewarn you of all the age-old traps that might lead to your downfall.

Take for instance this scheming pair that have been recently appearing side-by-side in your TYS:

  1. Find the range of values of x for which (1-3x)(x+2) ≤ 4.
  2. Find the range of values of p for which x2 – (p + 3)x + 2p + 3 > 0 for all values of x.

Note that one asks for the range of x while the other asks for the range of p. Miss Loi shall give it all away by revealing one requires you to find the discriminant while the other requires the roots. But both parts look so similar. So which is which?

Shouldn’t be a problem for the aristocrats and their grandmas, but for the rest of the grandmas out there the Quadratic Equations chapter can be a long and confusing one with all those sexy curves that naughty male students never fail to make jokes of. Tsk.

In any case, Miss Loi shall be nice by providing this chart that will hopefully refresh all grandmas’ ageing memories:

Discriminant
b2-4ac
f(x) = (xα)(xβ)
> 0Quadratic curve with positive discriminant
= 0Quadratic curve with zero discriminant
< 0Quadratic curve with negative discriminant

Now go for it Granny Power!

頑張って!!!

Revision Exercise

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Comments & Reactions

20 Comments

  1. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    3
    Wed
    6:21pm
     
    1

    1) 3< x < -2/3 ( < is greater or equal)

    2) cant do metally so lata>.>

  2. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    3
    Wed
    7:08pm
     
    2

    eh for 2nd i 4got abt the way discriminate should go>.

  3. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    3
    Wed
    7:09pm
     
    3

    gahh zz here comes the error

    uh b^2 - 4ac < 0

    (p-3) (p+1 ) < 0

    im guessing it should be
    3< p < -1

  4. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2007
    Oct
    4
    Thu
    2:15am
     
    4

    Kiroii,

    Miss Loi made a typo in Part 1 of the question (should be (1-3x)(x+2) ≤ 4). Just changed it, so could you retry this part (with workings) please? See, Miss Loi's also not immune to careless mistakes 😛

    For part 2, you're looking for the range of p which actually constitutes the coefficients of x2, x etc ... so this is the cue that you're gonna deal with the discriminant. Hence for the entire expression to be > 0 for all values of x (i.e. the entire curve must be above the x-axis), it's discriminant must be < 0, from the table given.

    So you're right and you managed to factorize and obtain the values of p at either end of the range.

    BUT YOUR FINAL ANSWER IS TRAGIC! How can any number be < -1 and > 3?????

  5. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    4
    Thu
    3:44am
     
    5

    juz coz you said please~

    after expanding

    -3x^2 - 5x +2 ≤ 4 ( < refers to greater or equalto) Next time try using &le; or &ge;
    0 ≤ 3x^2 + 5x + 2
    0 ≤ (3x + 2) (x+1)
    -2/3 ≤ x ≤ -1

    Yes you rightly looked for the roots of the expression when finding the range of x. You got your roots right but if you're confused for the range, it's always safe to quickly sketch out the curve for your obtained expression 0 ≤ (3x+2) (x+1) ≥ 0. Since the coefficient of x2 is positive, the curve should be a valley like this:

    Range of values of x

    You're looking for the range of values of x for which (3x+2) (x+1) ≥ 0. So from the diagram, it's clear you should be looking at the parts of the curve above the horizontal zero line (the shaded areas). So shouldn't your range be x ≤ -1 and x ≥ -2/3?

    as for 2 uh it's an imaginary curve yes? but i have never encountered a question where there are 2 values..usually 1 value..so i have no clue as to which way is which i was thinking
    +| - | + so>.

    Similarly for part 2, you're looking for the range of values of p for which (p-3) (p+1) < 0. So sketching out the curve again (once again it's a valley coz the coefficient of p2 is positive):

    Range of values of p

    This time you're looking for the range of values of p for which (p-3) (p+1) < 0. So from the diagram you should be looking at the part of the curve below the horizontal zero line (shaded area). So you range should be -1 < p < 3 ... which is logical.

    So to avoid careless mistakes and confusion when finding range of values in your exam, a simple quick sketch can do wonders in making things clearer. Just don't make wild guesses coz chances are you guesses will be wrong!

  6. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    4
    Thu
    3:51am
     
    6

    oh yea juz outta curiousity miss celine loi ya familar with history? o lvl syl of coz

  7. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2007
    Oct
    4
    Thu
    10:02am
     
    7

    Kiroii, just marked your workings and answers.

    History? As in the years of the different LV collections? 😀

  8. 123's Avatar
    123 commented in tuition class


    2007
    Oct
    4
    Thu
    6:35pm
     
    8

    i dun get it... For part 2, why is the discriminant less than 0?

  9. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2007
    Oct
    4
    Thu
    6:58pm
     
    9

    123,

    Read part 2 carefully. It says x2 - (p + 3)x + 2p + 3 > 0 for all values of x.

    What does this mean? It means the expression is always positive ⇒ the entire quadratic curve lies above the x-axis. And since it will never ever cut the x-axis ⇒ this curve has no real roots.

    And since the coefficient of x2 is positive, which scenario do you think this refers to in the table above which Miss Loi had painstakingly drawn up?

  10. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    4
    Thu
    11:07pm
     
    10

    eh but i dun get it why is an imaginary curve the area which is looking for is as you mentioned cant it be the same as part 1? and btw i was under the impression o lvls imaginary curves are rather elusive?
    and hist i dun mean LV collections>.< i ment "hail hitler" tat sorrta thingy anyway sent ya somemore questions-_-

  11. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2007
    Oct
    4
    Thu
    11:58pm
     
    11

    No it can't be the same as part 1.

    Let's summarize:

    In part 1, we have (3x + 2) (x+1) 0 so we're looking at the portion(s) of the curve above the zero line (i.e. x-axis).

    In part 2, we have (p-3) (p+1) < 0 so we're looking at the portion of the curve below the zero line.

    BTW Miss Loi is curious as to what's your definition of "imaginary curves"?

    As for history, Miss Loi can only "hail LV/Gucci/Prada", unfortunately.

  12. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    5
    Fri
    2:09am
     
    12

    uh RIGHT the < symbol ..
    er imaginary curves = discriminant
    it's whats written in my textbook >.< and i find it more apt then discriminant as this kinda curves are rather "stupid" ?

    as fer history>.

  13. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    5
    Fri
    2:10am
     
    13

    i din noe that tution teachers command salaries enough for such stuff?

  14. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2007
    Oct
    5
    Fri
    11:33am
     
    14

    Miss Loi only said "hail", as in "worship". Didn't say 'buy' 😛

  15. Prodigy's Avatar
    Prodigy commented in tuition class


    2007
    Oct
    5
    Fri
    6:51pm
     
    15

    Hi, kirroi. I am a fellow History Elective student too. What essay questions does your history teacher feels that will appear this year? My teacher has a sudden premonition that Japan/ Reasons for WWII will come out for section B, while China(1967-1989) will come our for Section A.

  16. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    5
    Fri
    11:59pm
     
    16

    huh reasons for ww2? dunno my teacher juz told us to focus on china, germany and russia esp germany comes out yearly but nonetheless i memorised the whole book btw im a pure hist student

  17. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    6
    Sat
    12:00am
     
    17

    btw china from 1967 ya mean china after mao? if im nt wrong smthin like china after ww2 came out fror a lvls last year

  18. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    6
    Sat
    12:02am
     
    18

    eh btw miss loi dun hail/ worship till becoming the idiom penny wise pound foolish by turning into a compulsive gucci must have tai tai~

  19. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2007
    Oct
    6
    Sat
    11:03pm
     
    19

    If you've read the About Miss Loi page, you'd find that she's already trying her very best in "preparing for her eventual tai tai lifestyle". But for the moment there's still a long way to go ...

    BTW guys, is there a Jφss Sticks for History somewhere on the net where you can discuss this?

  20. kiroii's Avatar
    kiroii commented in tuition class


    2007
    Oct
    6
    Sat
    11:20pm
     
    20

    >.< miss loi is kickin history aficionados out tats mean LOL

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