Factorise showing your work. 48a^2+4ab-140b^2
48a^2 + 4ab - 140b^2
Factor out 4
4(12a^2 + ab - 35b^2)
4(3a - 5b)(4a + 7b)
To factorize the expression 48a^2 + 4ab - 140b^2, we can use the method of factoring by grouping. Here's how it is done:
Step 1: Identify common factors, if any. In this case, we don't have any common factors among the terms.
Step 2: Look for pairs of terms whose coefficients have a common factor. In this case, the coefficients of the terms 48a^2 and -140b^2 are both divisible by 4.
Step 3: Take out the common factor from each pair and write it in front of parentheses. In this case, the common factor is 4, so we can write it outside the parentheses:
4(12a^2 + ab - 35b^2)
Step 4: Now, we need to factorize the expression within the parentheses: 12a^2 + ab - 35b^2. To do this, we look for two binomials that multiply together to give us this expression.
We need to find two numbers that multiply to give the constant term -35 (the product of -35 and the leading coefficient 12 is -420) and add up to the coefficient of the middle term (which is 1).
The numbers that satisfy this condition are -5 and 7:
12a^2 - 5ab + 7ab - 35b^2
Step 5: Group the terms accordingly:
(12a^2 - 5ab) + (7ab - 35b^2)
Step 6: Factor out the common factor from each grouping:
a(12a - 5b) + 7b(12a - 5b)
Step 7: Now, we can see that both terms have the factor (12a - 5b), so we can factor it out:
(12a - 5b)(a + 7b)
Thus, the fully factored form of the expression 48a^2 + 4ab - 140b^2 is 4(12a - 5b)(a + 7b).