Factorise each of the following expressions completely. (a) (1)/(4)a^(2)-ab-15b^(2)

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Factorise each of the following expressions completely. (a)  (1)/(4)a^(2)-ab-15b^(2)

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Identify Coefficients: We are given the quadratic expression (1/4)a^2 - ab - 15b^2, and we need to factor it completely. The first step is to identify the coefficients of the quadratic expression. The coefficient of a^2 is 1/4, the coefficient of ab is -1, and the coefficient of b^2 is -15.Find Multiplying Numbers: To factor the quadratic expression, we look for two numbers that multiply to give the product of the coefficient of a^2 and the coefficient of b^2, which is (1/4) * (-15) = -15/4, and add up to the coefficient of ab, which is -1.Use Correct Coefficients: We find that the numbers -5/2 and 3/2 satisfy these conditions because (-5/2) * (3/2) = -15/4 and (-5/2) + (3/2) = -1. These will be the coefficients of b in the factored form.Rewrite Middle Term: We rewrite the middle term -ab as the sum of two terms using the coefficients -5/2 and 3/2: (1/4)a^2 - (5/2)ab + (3/2)ab - 15b^2.Group Terms: Now we group the terms in pairs: [(1/4)a^2 - (5/2)ab] + [(3/2)ab - 15b^2].Factor by Grouping: We factor by grouping. For the first group, we can factor out (1/2)a, and for the second group, we can factor out 3b: (1/2)a[(1/2)a - 5b] + 3b[(1/2)a - 5b].Factor Common Factor: We notice that both groups contain the common factor of (1/2)a - 5b. We factor this out to get the final factored form: ((1/2)a - 5b)((1/2)a + 3b).

Factorise each of the following expressions completely. $\newline$ (a) $\frac{1}{4} a^{2}-a b-15 b^{2}$

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Factorise each of the following expressions completely. $\newline$ (a) $\frac{1}{4} a^{2}-a b-15 b^{2}$