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. x2=9x2=9
b. 2x298=02x298=0

Answer:
 (a)
x2=9x=±9x=±3x2=9x=±9x=±3

(b)
2x298=02x2=98x2=982=49x=±49=±72x298=02x2=98x2=982=49x=±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

Roots of Quadratic Equation – Example

Remember : Roots of a quadratic equation are the values of variables/unknowns that satisfy the equation.

Example
(a) Given x = 3 is the root of the quadratic equation x2+2x+p=0x2+2x+p=0 , find the value of p.
(b) The roots of the quadratic equation 3x2+hx+k=03x2+hx+k=0   are -2 and 4. Find the value of h and k.


General Form of Quadratic Equation

General Form of Quadratic Equation

General form of a quadratic equation is
ax2+bx+c=0ax2+bx+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) (3x5)2=0(3x5)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
  • 2x2+3x+4=02x2+3x+4=0
  • t2=25t2=25
  • y(6y3)=5y(6y3)=5
  • a3y+2y2=2a3y+2y2=2 where a is a constant.
Example of Non Quadratic Equation
  • 2x+1=02x+1=0 , (Reason: The highest power of x ≠ 2.)
  • 2x3+1=x2x3+1=x , (Reason: The highest power of x ≠ 2.)
  • t2+5t=3t2+5t=3 , (Reason: The present of the term 5t5t .)

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+2Let g1(x)=yg(y)=x3y+2=x        y=x23g1(x)=x23g(x)=3x+2Let g1(x)=yg(y)=x3y+2=x        y=x23g1(x)=x23

(a)(ii)
hg(x)=12x+5h(3x+2)=12x+5g(x)=3x+2Let u=3x+2   x=u23h(u)=12(u23)+5   =4u8+5   =4u3h(x)=4x3

(b)
gh(x)=g(4x3) =3(4x3)+2 =12x9+2 =12x712x7=8x+1   4x=8 x=2

Inverse Function Example 7


Example 7 (Comparison Method)
Given that f:x2hx3k , x≠3k , where h and k are constants and f1:x14+24xx , x≠0, find the value of h and of k.

Solution:




Inverse Function Example 6 (Comparison Method)


Example 6 (Comparison Method)
If f:xmxnx2,x2   and f1:x52x2x,x2. . Find the value of m and of n,


Inverse Function Example 5

Example 5

If g:xmxx3,x3 and g1(5)=14. Find the value of m.

 

Inverse Function Example 3


Example 3
Find the inverse function of f(x)=3x+25x+3