If the equation 3x^4+2x^3-6x^2-6x+p=0 has two equal roots, find the possible values of p.
x-1 is a factor so after replacing x=1 I got p=7. But I don't know any other possible values of p. I got p=1 but I solved it through inspection. Please help.
If the given information is that x-1 is a factor then f(1) = 0 ---> p = 7
A synthetic division by x-1 yields a quotient of (3x^3 + 5x^2 - x -7)
but another division by x-1 yields a quotient of (3x^2 + 8x +7)
so 3x^4+2x^3-6x^2-6x+p=0 ----> (x-1)(x-1)(3x^2 + 8x + 7) if p = 7
there's your double root.
The remaining quadratic yields two complex roots
To find the possible values of p, we need to use the fact that the equation has two equal roots.
If a quadratic equation has two equal roots, it means that the discriminant, which is the part inside the square root of the quadratic formula (b^2 - 4ac), is equal to zero.
In this case, we have a quartic equation. So, for the equation 3x^4 + 2x^3 - 6x^2 - 6x + p = 0 to have two equal roots, its discriminant (D) must be zero.
Let's start by finding the discriminant:
D = (2^2) - 4(3)(-6x^2 - 6x + p)
D = 4 + 72x^2 + 72x - 12p
For the equation to have two equal roots, D must be equal to zero.
Setting D equal to zero:
0 = 4 + 72x^2 + 72x - 12p
Rearranging and simplifying:
72x^2 + 72x - 12p = -4
Dividing both sides by 4:
18x^2 + 18x - 3p = -1
Now, for the equation to have two equal roots, the coefficients of the quadratic terms must be proportional. So, we can set up the equation:
18x^2 + 18x - 3p = -1
18(x^2 + x) - 3p = -1
Since we are looking for p, we can extract it from this equation:
3p = -1 - 18(x^2 + x)
3p = -1 - 18x^2 - 18x
Now, let's solve for p:
p = (-1 - 18x^2 - 18x) / 3
p = (-1 - 6x^2 - 6x) / 1
p = -1 - 6x^2 - 6x
From this equation, we can see that p is equal to -1 - 6x^2 - 6x for the equation 3x^4 + 2x^3 - 6x^2 - 6x + p = 0 to have two equal roots.
Therefore, there are infinitely many possible values of p that satisfy this condition. The value you found p=7 is one of them. Another possible value you mentioned, p=1, is not correct as it does not satisfy the condition of having two equal roots.