Question

Solve by factorisation method: (a) 8x² - 22x - 21= 0 (b) 3√5x² +25x +10√5= 0

Answer 1

Method to find solution of a quadratic equation by factorisation:

To find the solution of a quadratic equation by factorization method we use following  steps:

Write the given quadratic equation in standard form ax² + bx + c =0, and find the values of a, b and c.

Multiply a and c and split ac into prime factors. Then find two numbers p and q such that p-q = b  (if Sign of c is negative -ve).

Write the constant b of ax² + bx + c =0, as (p-q) and then factorise it.

Equate each factor to zero  and find the values of x.

SOLUTION:

a)Given quadratic equation is 8x² - 22x - 21= 0.
On comparing with standard form of quadratic equation i.e ax² + bx + c =0,

Here, a = 8 , b= -22 , c= -21
Now,ac= 8 × -21= -168

Here, c have negative sign so we have to find the factors of  168 whose difference is - 22. Such factors 28 and 6

8x² - 22x - 21= 0
8x² - 28x +6x  -21=0
4x (2x - 7) + 3(2x -7)= 0
(4x+3) ( 2x-7) = 0
(4x+3) = 0   or  ( 2x-7) = 0
4x = 0-3   or 2x= 0+7

x = -¾,    x = 7/2
Hence, roots of equation 8x² - 22x - 21= 0 are -¾ and 7/2.

b) Given quadratic equation is 3√5x² +25x +10√5= 0.
On comparing with standard form of quadratic equation i.e ax² + bx + c =0,
Here, a = 3√5 , b= 25, c= 10√5

Now,ac= 3√5 × 10√5= 30×5= 150

Here, c have +ve sign so we have to find the factors of  150  whose sum  is 25. Such factors 15 and 10.

3√5x² +25x +10√5= 0.
3√5x² +15x + 10x +10√5= 0.
3x(√5x + 5) + 2√5(√5x + 5)= 0
(3x +2√5) (√5x + 5) = 0
(3x +2√5) = 0   or  (√5x + 5) = 0
3x = -2√5     or √5x = -5

x = -2√5/3    or x = -5/ √5= -5×√5/√5×√5= -5√5/5= -√5  [by rationalising the denominator]

Hence, roots of equation 3√5x² +25x +10√5= 0 are -2√5/3 and -√5.

HOPE THIS WILL HELP YOU..

Answer 2

Hey!!

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Factoring  8x2-22x-21 

The first term is,  8x2  its coefficient is  8 .
The middle term is,  -22x  its coefficient is  -22 .
The last term, "the constant", is  -21 

Step-1 : Multiply the coefficient of the first term by the constant  
 8 • -21 = -168 

Step-2 : Find two factors of  -168  whose sum equals the coefficient of the middle term, which is   
-22 . -168   +   1   =   -167     -84   +   2   =   -82     -56   +   3   =   -53     -42   +   4   =   -38     -28   +   6   =   -22   That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -28  and  6 
                     8x2 - 28x + 6x - 21

Step-4 : Add up the first 2 terms, pulling out like factors :
                    4x • (2x-7)
Add up the last 2 terms, pulling out common factors :
                    3 • (2x-7)
Step-5 : Add up the four terms of step 4 :
                    (4x+3)  •  (2x-7)

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I Hope It's Help You...!!!