2 b b 4 ac x 2 a Chapter 5: Quadratic Equations in One - - PDF document

Chapter 5: Quadratic Equations in One Variable

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5.1 Quadratic Equations ةيعيبرتلا تلبداعملا

 An equation that contains one variable is said to be a quadratic equation if the highest power of the variable is 2.  This type of equations can be solved: (i) By factorisation )ةبرجتلاب( ليلحتلاب (ii) By using the quadratic formula ماعلا نوناقلاب (iii) By completing the square عبرملا لامكإب (iv) Graphically ينايبلا نسرلاب The general form of quadratic equation is: ax2 + bx + c = 0 Where a, b and c are constants.

SET 3

Chapter 5

Quadratic Equations in One Variable

لادحاو ريغتمب ةيعيبرتلا تلبداعم

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Chapter 5: Quadratic Equations in One Variable

5.2 Solving Quadratic Equations by Factorisation ليلحتلاب ةيعيبرتلا تلبداعملا لح

Example 1. Solve the equations: (a)

8 2

2

   x x and (b) 4 11 3

2

   x x

by factorisation. Solution:

(a)

8 2

2

   x x ) 2 )( 4 ( 8 2

2

     x x x x

The quadratic equation

8 2

2

   x x

becomes:

) 2 )( 4 (    x x

Either ) 4 (   x  4   x

• r

) 2 (   x  2  x So the roots of 8 2

2

   x x

are 4   x and 2 (b)

4 11 3

2

   x x ) 4 )( 1 3 ( 4 11 3

2

     x x x x

The quadratic equation

4 11 3

2

   x x

becomes:

) 4 )( 1 3 (    x x

Either ) 1 3 (   x  3 1   x

• r

) 4 (   x  4  x So the roots of 4 11 3

2

   x x

are 3 1   x and 4

5.3 Solving Quadratic Equations by Using the Quadratic Formula

ماعلا نوناقلاب ةيعيبرتلا تلبداعملا لح

 Using quadratic formula is a straightforward method for solving any quadratic

equation.

 The quadratic formula that is used to solve a quadratic equation having the form

• f is:

2

   c bx ax

a ac b b x 2 4

2 

  

Chapter 5: Quadratic Equations in One Variable

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Example 2: Solve

8 2

2

   x x

by using the quadratic formula. Solution: Comparing 8 2

2

   x x with

2

   c bx ax gives: a = 1, b = 2 and c = ‒ 8. Substituting these values into the quadratic formula

a ac b b x 2 4

2 

  

gives:

2(1) 8) 4(1)( 2 2

2

     x 2 6 2 2 36 2 2 32 4 2           x 2 6 2    x

• r

2 6 2   Therefore 2   2 4 x

• r

4    2 8

(as in example 1(a)).

Example 3: Solve

2 7 4

2

   x x

by using the quadratic formula. Solution: Comparing

2 7 4

2

   x x

with

2

   c bx ax

gives: a = 4, b = 7 and c = 2.

Substituting these values into the quadratic formula a ac b b x 2 4

2 

   gives: 2(4) ) 2 4(4)( 7 7

2 

   x 8 17 7 8 32 49 7        x

360 . 8 17 7      x

• r

390 . 1 8 17 7    

Hence the roots of

2 7 4

2

   x x

are

360 .   x

and

390 . 1   x

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Chapter 5: Quadratic Equations in One Variable

Example 4: Solve 2 . 4 3 . 10 68 . 2

2

   x x by using the quadratic formula. Solution: 68 . 2  a

,

3 . 10   b

and

2 . 4   c

.

Substituting the values of a, b and c into the quadratic formula

a ac b b x 2 4

2 

  

gives:

2(2.68) .2) 4 4(2.68)( ) 3 . 1 ( ) 3 . 1 (

2

       x 5.36 293 . 12 3 . 10 5.36 114 . 151 3 . 10 5.36 024 . 45 09 . 06 1 3 . 10        x 215 . 4 5.36 293 . 2 1 3 . 10    x

• r

372 . 5.36 293 . 2 1 3 . 10     x Thus, the roots of 4.2 10.3 2.68 2    x x are

215 . 4  x

and

372 .   x