if (SQRT(3)-1) is a root of the equation 2x^2 - 2kx+4=0 then k equals what.
2(√3 -1)^2 + 2k(√3-1)+4=0
divide by 2
(√3-1)^2 + (√3-1)k + 2 = 0
3-2√3+1 + (√3-1)k +2 = 0
(√3-1)k = 2√3-6
k = (2√3-6)/(√3-1) = -2√3
On the other hand, if you want rational coefficients, then if you think you might have a typo, and the equation is really
2x^2 - 2kx - 4 = 0
then you need the conjugate for a root, so divide by 2 and you have
x^2-kx-2 = 0
(x-(-1+√3))(x-(-1-√3)) = 0
((x-1)-√3)((x-1)+√3) = 0
(x-1)^2 - 3 = 0
x^2-2x+1-3 = 0
x^2-2x-2 = 0
2x^2-4x-4 = 0
and k=2
2x^2-2kx+4=0
2(sqrt3 -1)^2 + 2k(sqrt3-1)+4=0
2(2+ 3-2sqrt3)+2k(sqrt3-1)+4=0
k= (-14+4sqrt3)/(sqrt3 - 1) check that.
To find the value of k, we'll use the fact that if (SQRT(3)-1) is a root of the equation 2x^2 - 2kx + 4 = 0, then when we substitute x = SQRT(3) - 1 into the equation, the equation will equal zero.
So let's substitute x = SQRT(3) - 1 into the equation:
2x^2 - 2kx + 4 = 0
2(SQRT(3) - 1)^2 - 2k(SQRT(3) - 1) + 4 = 0
Expanding the equation:
2(3 - 2SQRT(3) + 1) - 2kSQRT(3) + 2k + 4 = 0
6 - 4SQRT(3) + 2 - 2kSQRT(3) + 2k + 4 = 0
Combining like terms:
12 - 4SQRT(3) - 2kSQRT(3) + 2k = 0
To find the value of k, we need to group the terms with SQRT(3) together and the constant terms together:
-4SQRT(3) - 2kSQRT(3) + 2k + 12 = 0
Now we can equate the coefficients:
-4SQRT(3) - 2kSQRT(3) = 0
-4 - 2k = 0
Simplifying the second equation:
-2k = 4
k = -2
Therefore, k equals -2.
To find the value of k, we can use the fact that the given expression (SQRT(3)-1) is a root of the equation 2x^2 - 2kx + 4 = 0.
Let's substitute the given root into the equation and solve for k.
Step 1: Substitute the root into the equation.
When x = SQRT(3)-1, the equation becomes:
2(SQRT(3)-1)^2 - 2k(SQRT(3)-1) + 4 = 0
Step 2: Simplify the equation.
Expanding the squared term, we get:
2(3 - 2SQRT(3) + 1) - 2k(SQRT(3)-1) + 4 = 0
6 - 4SQRT(3) + 2 - 2k(SQRT(3) - 1) + 4 = 0
12 - 4SQRT(3) -2k(SQRT(3) - 1) = 0
Step 3: Distribute the -2k term.
12 - 4SQRT(3) - 2kSQRT(3) + 2k = 0
Step 4: Group like terms.
(12 + 2k) - (4SQRT(3) + 2kSQRT(3)) = 0
Step 5: Factor out common terms.
2(6 + k) - 2SQRT(3)(2 + k) = 0
Step 6: Divide both sides by 2.
6 + k - SQRT(3)(2 + k) = 0
Step 7: Distribute the -SQRT(3) to each term inside the parentheses.
6 + k - 2SQRT(3) - kSQRT(3) = 0
Step 8: Combine like terms.
6 - 2SQRT(3) + k - kSQRT(3) = 0
Step 9: Rearrange the terms.
(6 + k) - (2SQRT(3) + kSQRT(3)) = 0
Step 10: Group like terms.
(6 + k)(1 - SQRT(3)) = 0
Step 11: Set each factor equal to zero and solve for k.
6 + k = 0 or 1 - SQRT(3) = 0
For the first factor:
k = -6
For the second factor:
1 - SQRT(3) ≠ 0 (since SQRT(3) is an irrational number)
Therefore, the value of k is -6.