factorise:-
(i). 6-x-2x^2
(ii).4y^2-8y+3
(iii).3m^2+24+36
To factorize the given expressions, we need to find the common factors or apply different methods depending on the type of expression.
(i). 6 - x - 2x^2:
For this expression, we can rearrange the terms to write it in descending order of powers of x:
-2x^2 - x + 6
To factorize, we look for two numbers that multiply to give us -12 (the coefficient of x^2 multiplied by the constant term), and also add up to the coefficient of x (-1).
The factors of -12 are either (-1, 12) or (1, -12), and we are looking for the pair that adds up to -1. Thus, the factorization is:
-2x^2 - x + 6 = (-2x + 3)(x - 2)
(ii). 4y^2 - 8y + 3:
For this expression, we can try to break down the middle term to factorize it easily. We look for two numbers that multiply to give us the product of the coefficient of y^2 (4) and the constant term (3), which is 12. We also want these numbers to add up to the coefficient of y (-8).
The factors of 12 that satisfy these conditions are (-6, -2). So, we rewrite the expression as:
4y^2 - 6y - 2y + 3
Now, we group the terms and find common factors:
(4y^2 - 6y) - (2y - 3)
2y(2y - 3) - 1(2y - 3)
Now we have a common term, (2y - 3), which we can factor out:
(2y - 3)(2y - 1)
(iii). 3m^2 + 24 + 36:
This expression can be simplified by combining like terms. The constant terms, 24 and 36, can be added to give us 60.
Therefore, the factorization of 3m^2 + 24 + 36 is:
3m^2 + 24 + 36 = 3m^2 + 60