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# roots - PowerPoint PPT Presentation

roots. roots. roots. What are the roots of a quadratic equations ?. roots. The Roots Of Quadratic Equations. Determine whether the x value given is satisfied for the quadratic equations respectively. 2x 2 + 5x + 3 = 0 ; x = -1 (b) x 2 – 8x + 16 = 0 ; x = 4

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roots

roots

roots

What are the roots

Determine whether the x value given is satisfied for the quadratic equations respectively.

• 2x2 + 5x + 3 = 0 ; x = -1
• (b)x2 – 8x + 16 = 0 ; x = 4
• (c ) x(9 – 2x) + 2 = 8 ; x = - 2
• (d)(x + 2)(x + 3) = 20 ; x = 7

2(-1)2 + 5(-1) + 3 = 2 – 5 + 3 = 0

yes

(4)2 – 8(4) + 16 = 16 – 32 + 16 = 0

yes

-2(9 + 4) + 2 = -26 + 2 = - 24

no

(7 + 2)(7 + 3) = 9 x 10 = 90

no

equation are the values

of the unknown which

equations.

Determine the roots of quadratic equations from the factorisation formed

(ax + b) ( cx + d) = 0

Step 1

ax + b = 0 or cx + d = 0

Step 2

x =

x =

or

the roots

Determine the roots of quadratic equations in form of

(ax + b) ( cx + d) = 0

Examples

(a) (3y – 2)(y + 4) = 0 (b) 2x (x – 3) = 0

Solutions

(a) (3y – 2)(y + 4) = 0

• 2x (x – 3) = 0
• 2x = 0
• x = 0
• or x – 3 = 0
• x = 3

3y – 2 = 0

y =

or

y + 4 = 0

y = - 4

example 2

thinking process

a) solve the equations

(a)(i) subtract 12t from both sides of equation, factorise and solve for t

(i)5t2 = 12t

do not divide

both sides

of equation

by t. you may

lose one possible

value of t.

5t2 = 12t

5t2 = - 12t = 0

t(5t – 12) = 0

t = 0 @

5t – 12 = 0

t =

(ii)

b) Solve the equation

3x2 + 9x + 5 = 0

b) The quadratic expression is not factorisable

FACTORISATION

(3 TERMS)

METHOD 1 : CROSS - MULTIPLICATION

FACTORISED

x2 + 2x – 15 = 0

-3x

x - 3

x + 5

= 0

x - 3

5x +

Then

x + 5

x - 3

= 0

x = 3

x2

-15

2x

or

x + 5

= 0

The roots are

x = 3 @ -5

x = - 5

FACTORISATION

(3 TERMS)

METHOD 2 :

1 X 6

x2 + 5x + 6=0

choose 2x 3

x2 + 5x + 6

2 X 3

(+2x) +(+ 3x) = 5x

x2 + 2x + 3x + 6

5x is changed to 2x + 3x

(x2 + 2x) + (3x + 6)

x(x + 2) +3(x + 2)

factorise

(x + 2)(x + 3) =0

Factorise again

x + 2 = 0 or x + 3 = 0

x = - 2 or - 3

Two roots

factorisation

METHOD 3 :

This abstract way of factorisation can be explained in geometry form known as

Dienes’ Algebraic Experience Material

Example of factorisation

(3 terms )

No of big Square 1

No of rectangle 5

No. of small squares 6

example 1 :

x2 + 5x + 6

The arrangement:

These cards are to be arranged to form a rectangle

(x + 3)

(x+2)

(x + 3 )(x + 2) = 0

x = - 3 or - 2

The roots:

closure

equation are the values

of unknown

which satisfied

ax2 + bx + c = 0

common form

no

factorisation

yes

(ax + b)(cx + d) = 0

another

methods

ax + b = 0 or cx + d = 0

roots

x =

x =

or