SPM Add Math Form 4 Chapter 2 Quadratic Equations

SPM Add Math Form 4 Chapter 2 Quadratic Equations CHAPTER 2 : QUADRATIC EQUATIONS

General Form of Quadratic Equations and the Roots of Quadratic Equation Example 1:

Given that 2 is one of the roots of the quadratic equation x 2  kx  4  0, find the positive value of k.

Solving Quadratic Equation (i)

Factorisation

Example 2 :

Find the roots of the following quadratic equations using factorisation : (a) 4x 2  18 x  0 (b) 2x 2  7 x  6  0 Example 3 :

Determine the roots of the quadratic equations below : (a) 3x(x - 1)  8 - x 5 (b) 4x   12 x

(ii)

Completing the Square

Example 4: By completing the square, solve the following quadratic equations :

(a) x 2  4 x  7  0 (b) 2x 2  12 x  9  0

(iii)

Quadratic formula

Example 5 : Solve the following quadratic equations using the quadratic formula :

Copyright www.epitomeofsuccess.com

Page 1

SPM Add Math Form 4 Chapter 2 Quadratic Equations (a) 2x 2  6 x  3  0 (b) 4x 

2 9 x

For the Complete Worksheet and Answers, please join as a member 1 www.epitomeonlinetuition.com. Writeat the quadratic equation with the roots 6, 3

Forming the Quadratic Equations Example 6:

Example 7:

4 When you join as anda 5 Bronze or 3 Silver Member, you will gain unlimited the Quadratic equation 2x  x  access 6  0 has  and  to as its all roots with   . (a) Find the values of  and  . Worksheets and Video Tutorials for (b) Form the equation with the roots of (i)  , SPM Add Math both Form 4 and (ii) 2  1, 3 - 2 Form 5 Write the quadratic equation given the roots -

Example 8 :

2

2

2

Example 9 :

Quadratic equation 3x 2  6 x  8  0 has the roots of  and  . Form the new quadratic equation with the roots of (a)   1,   1 (b)

1



,

1



Determining the unknown values in the quadratic equation Example 10 :

Given the roots of the quadratic equation x 2  (h  2) x  6k  0 are 3 and - 4. Find the values of h and k. Example 11 :

Given that the roots of the quadratic equation x 2  (q  1) x  2 are 2 and p. Calculate the values of p and q. Copyright www.epitomeofsuccess.com

Page 2

SPM Add Math Form 4 Chapter 2 Quadratic Equations Example 12 :

Given that the difference between the roots of the quadratic equation x 2  px  q  0 is 2. Show that p 2  4q  4. Example 13 :

If one of the roots of the quadratic equation 2x 2  mx  9  0 is twice the value of the other root, find

For the Complete Worksheet and Answers, please join as a member at www.epitomeonlinetuition.com.   Equation 3x  ( p  3) x  2q  0 has the roots of and with     5 and   6.

(a) the possible values of m

(b) the possible values of the roots Example 14 :

2

3

3

Find the values of p and q.

When you join as a Bronze or Quadratic equation 2x Member,  kx  20  0 has the roots of  and  with gain    and  -   3. Silver you will Determine access to all the (a) the unlimited values of  and  . (b) the possible values of k. Worksheets and Video Tutorials for SPM Add Math both Form 4 and Form 5

Example 15 :

2

Usage of Discriminant

1) Determining the nature of the roots of a quadratic equation Example 16 : Determine the type of roots for the quadratic equations given below:

(a) x 2  8 x  14  0 (b) 3x 2  7 x  5  0

2) Determining the value of the range of values of unknown in the equation given the nature of roots

Example 17 :

Copyright www.epitomeofsuccess.com

Page 3

SPM Add Math Form 4 Chapter 2 Quadratic Equations Determine the range of values of p if quadratic equation x 2  4 x  2 p  0 has two different roots. Example 18 :

If quadratic equation x 2  2( p  3) x  p  1  0 has only one root, find the values of p.

For the Complete Worksheet and Find the range of values of k if quadratic equation (k  5)x  8x  1  0 has no real roots. Answers, please join as a member at www.epitomeonlinetuition.com.

Example 19 :

2

Example 20 :

Given that the quadratic equation x 2  2kx  3k  0 has real roots. Determine the range of values of k.

When you join as a Bronze or Silver Member, you will gain unlimited access to all the and Video Tutorials for (a)Worksheets y  x  4x  5 6 (b) yxSPM Add Math both Form 4 and x Form 5 3) Determining the type of intersection between a straight line and a curve

Example 21:

Determine whether the straight line y=2x+1 intersects, touches or does not intersect with the curves given as below: 2

Example 22 :

Determine the range of values of k if the straight line y  4 - kx intersects the curve y  x 2  4 x  5 at two points. Example 23:

If the straight line y  3x  k is a tangent to the curve y  4x 2  5x  2, find the value of k. Example 24:

Determine the range of values of m given that the straight line y  mx  1 does not intersect with the curve y 2  8 x.

Copyright www.epitomeofsuccess.com

Page 4