Quadratic Equations (for IGCSE 10 / Form 4 Students)

Hello IGCSE Yr 10 or 11 / Forms 4 or 5 Students,

1) Taking Add. Maths for your SPM 2012 / IGCSE 2013 exams and beyond? Yes? Then this blog should interest you. Btw I'm Tan residing in PJ, Selangor, Malaysia and can be reached at: tutortan1@gmail.com or SMS 011-2328 7882.

2) One Form 4 student (who took tuition from me) told me her first encounter with Add Maths was "Quadratic Equations (QE)" (F4 Chapter 2). Thereafter, she learned "Simultaneous Equations" (F4 Chapter 4) and now about to learn "Functions" (F4 Chapter 1). Other schools may start differently...

3) After going through the QE chapter with her, I posed 20 questions to test her. Among which is Q 12 which requires her to derive the "Quadratic Root Formula". It's simple - and I have taught her how to (surprisingly, her F4 textbook omits to show them how to - by "completing the square" method!). Why don't you try Q 12 as well if not all the questions (some drafted by me, others from F4 textbook).

4) Maths is best learnt by understanding rather than by rote learning. I believe if you know how to derive the "Quadratic Root Formula", you need not memorise it. You can rewrite it anytime you wish. Also once you know how to derive it, the Discriminant, "b² – 4ac" will no longer be so alien and distant to you.

5) Another thing is the general name given to that formula - it's known as "Quadratic Formula" instead of "Quadratic Root Formula". The latter name should be used as it is more specific - it is not just any quadratic formula but a quadratic formula for finding the roots of a quadratic equation: Hence, I prefer the term "Quadratic Root Formula" which is more friendly to newbies in Add. Maths.

Happy doing maths...

Form 4 Chapter 2: Quadratic Equations

Tutorial Questions:

1. In your own words, describe a quadratic algebraic expression.

2. Explain why x^2 + 2/x – 3 = 0 is not a quadratic equation.

3. Is x^3 – x^2 = 0 a quadratic equation?

4. By substitution, determine whether x = 1, x = 2 and/or x = -2 are roots of x^2 + x - 2 = 0

5. By inspection, determine whether x = 2, x = -2, x = 3 and/or x = -3 are roots of the equation (x – 2)(x + 3) = 0

6. By inspection, state the roots of the equation (x – p)(x – q) = 0

7. The table below gives the values of the dependent variable y of a quadratic equation: y = x^2 – 2x – 15 in terms of the independent variable x.

x	6	4	2	0	-2	-4
y	9	-7	-15	-15	-7	9

1. By inspection, identify the locations of the roots of the quadratic equation as between which pair(s) of x values?
2. By trial and improvement method, find the roots of the quadratic equation.

8. Find the roots of the following quadratic equations by factorization:
1. x^2 + x -12 = 0
2. 12x^2 = 6 – x
3. x(2x – 5) = 12

9. You can make any quadratic expression in the form of x^2 + bx into a perfect square in what is known as ‘completing the square’. Show me how do you complete x^2 + bx into a perfect square.

10. For the quadratic equation x^2 + 6x – 1 = 0:
1. State whether you can easily factorise it into the form: (x – p)(x – q) = 0;
2. Find its roots by “completing the square” method instead of factorization.

11. Solve the quadratic equation 2x^2 + 4x -3 = 0 by completing the square.

12. Find the roots of the quadratic equation ax^2 + bx + c = 0, where a is not equal to 0, by completing the square.  

13. Use the quadratic roots formula you derive in Q12 above to find the solutions of the equation 10x^2 + 3x = 16. Give your answers correct to 4 significant numbers.

14. Q12 shows that you can find the roots of a quadratic equation by the derived quadratic root formula.
1. What is “b^2 – 4ac” known as?
2. If b^2 – 4ac > 0, what can say about the roots of the quadratic equation?
3. If b^2 – 4ac = 0, what can say about the roots of the quadratic equation?
4. If b^2 – 4ac < 0, what can say about the roots of the quadratic equation?

15.  For the quadratic equation (k +1)x^2 – 4x + 9 = 0, find:
1. The value of k, if the equation has 2 equal roots
2. The range of k, if the equation has 2 different roots
3. The range of k, if the equation has no real roots

16. For the quadratic equation x^2 – 4mx + n^2 = 0, derive the algebraic relations between m and n when the equation has:
1. 2 equal roots
2. 2 different roots
3. no real roots

17. Q6 above shows that the quadratic equation (x – p)(x – q) = 0 has the roots p and q.
1. Expand (x – p)(x – q) = 0
2. Show from Q17a that if the roots of a quadratic equation are known, the quadratic equation can be formed from the given roots as:

x^2 –(sum of roots)x + (product of roots) = 0

18. Form a quadratic equation with the roots 2 and -3.

19. Given that a and b are the roots of the quadratic equation:
1. 2x^2 + 2x – 5 + 0, form a quadratic equation with the roots (a – 2) and (b – 2)
2. 2x^2 + 3x – 9 = 0, form a quadratic equation with the roots

                                                               i.      a/2 and b/2

                                                             ii.      3a and 3b

20. If one of the roots of the equation 2x^2 + px +9 = 0 is twice the other root, find the possible values of p.