Given that the roots of the equation 3x^2+x+2=0 are a and b, show that 27a^4=11a+10.
To show that 27a^4 = 11a + 10, we can start by finding the sum and product of the roots (a and b) using Vieta's formulas.
For a quadratic equation of the form ax^2 + bx + c = 0, Vieta's formulas state that:
1. The sum of the roots (a + b) is equal to -b/a.
2. The product of the roots (ab) is equal to c/a.
In our given equation 3x^2 + x + 2 = 0, we have a = a and b = b. Comparing it with the general form ax^2 + bx + c = 0, we can see that a = 3, b = 1, and c = 2.
So, the sum of the roots (a + b) is equal to -b/a:
a + b = -b/a
a + b = -1/3 ----(1)
The product of the roots (ab) is equal to c/a:
ab = c/a
ab = 2/3 ----(2)
Now, let's use the values of a and b to express the equation 27a^4 = 11a + 10:
27a^4 - 11a - 10 = 0.
To simplify this equation, let's express it in terms of a and b using Vieta's formulas:
The sum of the product of the roots taken two at a time (ab)^2 can be expressed as (a + b)^2 - 2ab:
(ab)^2 = (a + b)^2 - 2ab.
Using the values from equations (1) and (2), we can substitute:
(ab)^2 = (-1/3)^2 - 2(2/3)
(ab)^2 = 1/9 - 4/3
(ab)^2 = 1/9 - 12/9
(ab)^2 = -11/9.
Now, let's solve for (ab)^2:
(ab)^2 = -11/9
ab = √(-11/9).
Since we know that ab = 2/3 from equation (2), we can write:
2/3 = √(-11/9).
To simplify, we can square both sides:
(2/3)^2 = (√(-11/9))^2
4/9 = -11/9.
As 4/9 does not equal -11/9, this implies that the equation 27a^4 = 11a + 10 does not hold true for the roots of the given quadratic equation 3x^2 + x + 2 = 0.
To show that 27a^4 = 11a + 10, we need to use the fact that the roots of the given quadratic equation 3x^2 + x + 2 = 0 are a and b.
Since a and b are the roots, we can write the equation in factored form as (x - a)(x - b) = 0. Expanding this equation, we get:
x^2 - (a + b)x + ab = 0
Comparing this equation with the given quadratic equation 3x^2 + x + 2 = 0, we can equate the coefficients:
Coefficient of x^2:
1 = 3 (which is the same)
Coefficient of x:
-(a + b) = 1
Coefficient of constant term:
ab = 2
We have a system of equations to find the values of a and b.
From the equation -(a + b) = 1, we can rewrite it as a + b = -1.
Now, let's square the equation a + b = -1 to eliminate the variables a and b.
(a + b)^2 = (-1)^2
Expanding the square on the left side of the equation, we get:
a^2 + 2ab + b^2 = 1
Since we already know that ab = 2, we can substitute it into the equation:
a^2 + 2(2) + b^2 = 1
Simplifying further:
a^2 + 4 + b^2 = 1
Rearranging the terms:
a^2 + b^2 = 1 - 4
a^2 + b^2 = -3
Now, let's square the equation a^2 + b^2 = -3 to eliminate the variables a and b again.
(a^2 + b^2)^2 = (-3)^2
Expanding the square on the left side of the equation, we get:
a^4 + 2a^2b^2 + b^4 = 9
Since we already know that a^2b^2 = 4, we can substitute it into the equation:
a^4 + 2(4) + b^4 = 9
Simplifying further:
a^4 + 8 + b^4 = 9
Subtracting 8 from both sides:
a^4 + b^4 = 1
Now, let's multiply the equation a + b = -1 with a^3 + b^3:
(a + b)(a^3 + b^3) = -1(a^3 + b^3)
Using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2), we can rewrite the equation:
(a + b)(a^2 - ab + b^2) = -1(a^3 + b^3)
Substituting the known values, we get:
(-1)(a^2 - ab + b^2) = -1(a^3 + b^3)
Simplifying further:
ab - a^2 - b^2 = a^3 + b^3
Substituting the values of ab, a^2, and b^2 we found earlier:
2 - (a^2 + b^2) = a^3 + b^3
Since a^2 + b^2 = -3, we can substitute it into the equation:
2 - (-3) = a^3 + b^3
Simplifying further:
2 + 3 = a^3 + b^3
5 = a^3 + b^3
Now, let's raise the equation a + b = -1 to the power of 3:
(a + b)^3 = (-1)^3
Using the formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, we can rewrite the equation:
a^3 + 3a^2b + 3ab^2 + b^3 = -1
Substituting the known values, we get:
a^3 + 3a^2b + 3ab^2 + b^3 = -1
Substituting the values of ab, a^2, and b^2 we found earlier:
a^3 + 3(2) + 3(2) + b^3 = -1
Simplifying further:
a^3 + 6ab + 6ab + b^3 = -1
Since we already know that 6ab = -6, we can substitute it into the equation:
a^3 - 6 + 6 + b^3 = -1
Simplifying further:
a^3 + b^3 = -1
Comparing this equation with the previous equation we found, a^3 + b^3 = 5, we can see that there is a contradiction.
Hence, the statement 27a^4 = 11a + 10 is not true for the given roots a and b of the equation 3x^2 + x + 2 = 0.
Using brute force, since a = (-1+√23 i)/6
27a^4 = (49+11√23 i)/6
and so does 11a+10