The roots of the equation 3x²-2x-4=0 are J and K. Evaluate J² + K².

28/9

Step-by-step explanation:

If the roots are J and K, then:

3 (x - J) (x - K) = 0

3 (x² - (J+K) x + JK) = 0

So if we factor out the leading coefficient:

3x² - 2x - 4 = 0

3 (x² - 2/3x - 4/3) = 0

The coefficient of the second term is the sum of the roots:

J + K = 2/3

And the constant is the product of the roots:

JK = - 4/3

If we take the sum of the roots and square it:

(J + K) ² = (2/3) ²

J² + 2JK + K² = 4/9

And subtract twice the product:

J² + K² = 4/9 - 2JK

J² + K² = 4/9 - 2 (-4/3)

J² + K² = 4/9 + 8/3

J² + k² = 28/9