Factorize the algebraic expression
a) x^2-3x+18
b) y^2+y=20
c) x^2-2x-8
d) y^2+3y+28
a) To factorize the expression x^2-3x+18, we need to find two numbers that multiply to give 18 (the coefficient of the constant term) and add up to -3 (the coefficient of the x term). In this case, the numbers are -6 and -3. Therefore, we can rewrite the expression as (x-6)(x-3).
b) To factorize the expression y^2+y=20, we need to rearrange it to the form y^2+y-20=0. Now we need to find two numbers that multiply to give -20 (the product of the coefficient of the y^2 term and the constant term) and add up to +1 (the coefficient of the y term). In this case, the numbers are +5 and -4. Therefore, we can rewrite the expression as (y+5)(y-4).
c) To factorize the expression x^2-2x-8, we need to find two numbers that multiply to give -8 (the product of the coefficient of the constant term) and add up to -2 (the coefficient of the x term). In this case, the numbers are -4 and +2. Therefore, we can rewrite the expression as (x-4)(x+2).
d) To factorize the expression y^2+3y+28, we need to find two numbers that multiply to give 28 (the product of the coefficient of the constant term) and add up to +3 (the coefficient of the y term). However, there are no two such numbers that satisfy these conditions. Therefore, the expression y^2+3y+28 is not factorizable over the set of real numbers.