2(a). To find the equation whose roots are a^2 and b^2, we can use the fact that if a and b are roots of an equation, then x = a and x = b satisfy that equation.
Since alpha and beta are the roots of 2x^2-7x+4=0, we have:
alpha + beta = 7/2 (sum of roots)
alpha * beta = 4/2 = 2 (product of roots)
Substituting x = a^2 and x = b^2, we can find the equation with roots a^2 and b^2:
(a^2) + (b^2) = 7/2
(a^2) * (b^2) = 2
Expanding the equation and simplifying, we get:
(a^2 + b^2)^2 - 2(a^2)(b^2) = (7/2)^2 - 2(2)
a^4 + 2(a^2)(b^2) + b^4 - 4 = 49/4 - 4/1
a^4 + 2(a^2)(b^2) + b^4 - 4 = 37/4
Therefore, the equation whose roots are a^2 and b^2 is:
x^2 + 2(a^2)(b^2) + (b^2)^2 - 4 = 37/4.
2(b). We can simplify the expression (2ab+b)^2 - (b-2a)^2 without using a mathematical table or calculator.
(2ab+b)^2 - (b-2a)^2 = (2ab+b)^2 - (b^2 - 4ab + 4a^2)
Expanding both terms, we get:
(4a^2b^2 + 4ab + b^2) - (b^2 - 4ab + 4a^2)
Simplifying, we cancel out like terms:
4a^2b^2 + 4ab + b^2 - b^2 + 4ab - 4a^2
Combining like terms, we have:
4a^2b^2 + 8ab - 4a^2
Therefore, without using a mathematical table or calculator, (2ab+b)^2 - (b-2a)^2 equals 4a^2b^2 + 8ab - 4a^2.
2(a). If alpha and beta are the root of the equation 2x^2-7x+4=0 find the equation whose roots are a^2 and b^2
2(b). Without using mathematical table or calculator, evaluate (2ab+b)^2 - (b-2a)^2
1 answer