Final answer:
The quadratic equation with roots (-1 - sqrt(2))/3 and (-1 + sqrt(2))/3 and leading coefficient of -3/5 is -54/5x² - 18/5x + 1/5 = 0.
Step-by-step explanation:
To find the quadratic equation with given roots, we can use the fact that if α and β are roots of the equation, then the equation can be written as:
a(x - α)(x - β) = 0
Given the roots α = (-1 - √2)/3 and β = (-1 + √2)/3 and the leading coefficient -3/5, we can substitute into the equation:
-3/5(x + (1 + √2)/3)(x + (1 - √2)/3)
Expanding this will give us the quadratic equation. We multiply the factors first:
(x + (1 + √2)/3)(x + (1 - √2)/3) = x² + x(1 - √2)/3 + x(1 + √2)/3 + (1 - √2)(1 + √2)/9
Combining like terms and multiplying by 9 to clear the fraction gives us:
9x² + 3x - (1 - 2)/3 = 9x² + 3x - 1/3
Finally, multiply by -3/5:
-3/5 * (9x² + 3x - 1/3) = -54/5x² - 18/5x + 1/
The quadratic equation with the given roots and leading coefficient is therefore:
-54/5x² - 18/5x + 1/5 = 0