Given that the roots of 2x^2-4x+5 are alpha and beta,find the value of ...
(i)alpha^3beta^3
(ii)(alpha-beta)^2
Using the sum and product of roots property, we know
a+b = - (-4/2) = 2 and ab = 5/2, where a and b are alpha and beta
we know that
(a+b)^3 = a^3 + b^3 + 3a^2b + 3ab^2
(a+b)^3 = a^3 + b^3 + 3ab(b+a)
2^3 = a^3 + b^3 + 3(5/2)(2)
8 = a^3 + b^3 + 15
a^3 + b^3 = -7
in the same way....
(a-b)^2 = a^2 - 2ab + b^2
= a^2 +b^2 - 2ab
but a^2 + b^2 = (a+b)^2 - 2ab
so
(a=b)^2 = (a+b)^2 - 2ab - 2ab
= 4 - 5/2 - 5/2
= -1
in the third last line I should have had
(a-b)^2 = (a+b)^2 - 2ab - 2ab
To find the values of alpha^3 * beta^3 and (alpha - beta)^2, we need first to determine the values of alpha and beta.
For a quadratic equation in the form of ax^2 + bx + c = 0, the roots can be found using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, we have the quadratic equation 2x^2 - 4x + 5 = 0. Comparing it to the general form ax^2 + bx + c = 0, we have a = 2, b = -4, and c = 5.
Using the quadratic formula, we can calculate the roots:
x = (-(-4) ± √((-4)^2 - 4 * 2 * 5)) / (2 * 2)
x = (4 ± √(16 - 40)) / 4
x = (4 ± √(-24)) / 4
Since the discriminant (-24) is negative, the roots will be complex numbers. Let's simplify the expression:
x = (4 ± 2√6i) / 4
x = 1 ± √6i
So, the values of alpha and beta are 1 + √6i and 1 - √6i, respectively.
Now let's solve for the values of alpha^3 * beta^3 and (alpha - beta)^2:
(i) alpha^3 * beta^3
To simplify this expression, we substitute the known values:
alpha^3 * beta^3 = (1 + √6i)^3 * (1 - √6i)^3
Expanding each binomial term, we get:
alpha^3 * beta^3 = (1 + 3√6i + 3(√6i)^2 + (√6i)^3) * (1 - 3√6i + 3(√6i)^2 - (√6i)^3)
Simplifying further:
alpha^3 * beta^3 = (1 + 3√6i - 18 - 6√6i) * (1 - 3√6i - 18 + 6√6i)
alpha^3 * beta^3 = (-17 - 3√6i) * (-17 + 3√6i)
Multiplying the terms:
alpha^3 * beta^3 = 17^2 - (3√6i)^2
alpha^3 * beta^3 = 289 - 54i^2
Since i^2 = -1, we can simplify further:
alpha^3 * beta^3 = 289 - 54(-1)
alpha^3 * beta^3 = 289 + 54
alpha^3 * beta^3 = 343
So, alpha^3 * beta^3 is equal to 343.
(ii) (alpha - beta)^2
Again, substituting the known values:
(alpha - beta)^2 = (1 + √6i - (1 - √6i))^2
(alpha - beta)^2 = (2√6i)^2
(alpha - beta)^2 = (2^2)(√6i)^2
(alpha - beta)^2 = 4 * (-6)
(alpha - beta)^2 = -24
So, (alpha - beta)^2 is equal to -24.
Thus, the values of alpha^3 * beta^3 and (alpha - beta)^2 are 343 and -24, respectively.