2.2.2c Solving Quadratic Equations – Quadratic Formula

The quadratic equation a x 2 + b x + c = 0 can be solved by using the quadratic formula

b ± b 2 4 a c 2 a

Example 
Use the quadratic formula to find the solutions of the following equations.
a. x 2 + 5 x 24 = 0
b. x ( x + 4 ) = 10

Answer
(a)
For the equation x 2 + 5 x 24 = 0
a = 1, b = 5, c = -24

x = b ± b 2 4 a c 2 a x = ( 5 ) ± ( 5 ) 2 4 ( 1 ) ( 24 ) 2 ( 1 ) x = 5 ± 121 2 x = 8  or  x = 3

(b)
x ( x +  4 )   =  10 x 2 + 4 x 10 = 0 a = 1 ,    b = 4 ,    c = 10 x = b ± b 2 4 a c 2 a x = ( 4 ) ± ( 4 ) 2 4 ( 1 ) ( 10 ) 2 ( 1 ) x = 4 ± 56 2 x = 1.742  or  x = 5.742

2.2b Solving Quadratic Equations – Completing the Square (Examples)


2.4.2a Solving Quadratic Equations – Completing the Square (Examples)

(A) Steps to solve quadratic equation using completing the square
make sure that the coefficient of x is 1.
Rewrite the equation 
ax2 + bx + c = 0 in the form ax2 + bx = –c.
Add ( coefficient of  x 2 ) 2  to both side of the equation.

Example:
Solve the following quadratic equations by completing the square.
(a) x2 6x 3 = 0
(b) 2x2 5x 7 = 0
(c) x2 + 1 = 10x/3 

Solution:









2.2b Solving Quadratic Equations – Completing the Square

2.4.2 Solving Quadratic Equations – Completing the Square

(A) The Perfect Square
1. The expression x2 + 2x + 1 can be written in the form (x + 1)2, it is called a “perfect square”.

2. If the algebraic expression on the left hand side of the quadratic equation is a perfect, the roots can be easily obtained by finding the square roots.

Example:
Solve each of the following quadratic equation
(a) (x + 1)2 = 25
(b) x2 8x + 16 = 49

Solution:
(a)
(x + 1)2 = 25
(x + 1)2 = ±√25
x = 1 ± 5
x = 5  or  x = 6

(b)
x2 8x + 16 = 49
(x 4)2 = 49
(x 4) = ±√49
x = 4 ± 7
x = 11  or  x = 3


(B) Solving Quadratic Equation by Completing the Square

1. To solve quadratic equation, we make the left hand side of the equation a perfect square.

2. To make any quadratic expression x2 + px into a perfect square, we add the term (p/2)2 to the expression and this will make 





3. The following shows the steps to solve the equation by using completing the square method for quadratic equation ax2+ bx = – c.
 (a) Rewrite the equation in the form ax2 + bx = – c.
 (b) If the coefficient a ≠ 1, reduce it to 1 (by dividing).
 (c) Add (p/2)2 to both sides of the equation.
 (d) Write the expression on the left hand side as a perfect square.
 (e) Solve the equation.

Solving Quadratic Equations

Solving Quadratic Equations

To solve a quadratic equation means to find all the roots of the quadratic equations.

Example:
Find the roots of the quadratic equations
a. x 2 = 9
b. 2 x 2 98 = 0

Answer:
 (a)
x 2 = 9 x = ± 9 x = ± 3

(b)
2 x 2 98 = 0 2 x 2 = 98 x 2 = 98 2 = 49 x = ± 49 = ± 7

A quadratic equation may be solve by using one of the following method
  1. Factorisation
  2. Completing the square
  3. Using quadratic formula

General Form of Quadratic Equation

General Form of Quadratic Equation

General form of a quadratic equation is
a x 2 + b x + c = 0
where a, b, and c are constants and a≠0.

*Note that the highest power of an unknown of a quadratic equation is 2.


Example (Find the values of a, b and c)
Rewrite the following into the general form of a quadratic equation. Find the values of a, b, and c.
(a) ( 3 x 5 ) 2 = 0
(b) (x - 8)(x + 8) = 0

Solution





Quadratic Equations (Introduction)

What is a Quadratic Equation?

  1. A quadratic equation is a polynomial equation of the second degree. 
  2. A quadratic equation has only one variable 
  3. The highest power of the variable is 2.
Example of Quadratic Equation
The followings are some examples of quadratic equations
  • 2 x 2 + 3 x + 4 = 0
  • t 2 = 2 5
  • y ( 6 y 3 ) = 5
  • a 3 y + 2 y 2 = 2 where a is a constant.
Example of Non Quadratic Equation
  • 2 x + 1 = 0 , (Reason: The highest power of x ≠ 2.)
  • 2 x 3 + 1 = x , (Reason: The highest power of x ≠ 2.)
  • t 2 + 5 t = 3 , (Reason: The present of the term 5 t .)

SPM Practice (Long Question)


Question 5:
In diagram below, the function g maps set P to set Q and the function h maps set Q to set R.



Find
(a) in terms of x, the function
(i) which maps set Q to set P,
(ii) h(x).
(b) the value of x such that gh(x) = 8x + 1.


Solution:
(a)(i)
g( x )=3x+2 Let  g 1 ( x )=y g( y )=x 3y+2=x         y= x2 3 g 1 ( x )= x2 3

(a)(ii)
hg( x )=12x+5 h( 3x+2 )=12x+5 g( x )=3x+2 Let u=3x+2    x= u2 3 h( u )=12( u2 3 )+5    =4u8+5    =4u3 h( x )=4x3

(b)
gh( x )=g( 4x3 )  =3( 4x3 )+2  =12x9+2  =12x7 12x7=8x+1    4x=8  x=2