Find all the roots α, β, γ of the cubicequation 3 7 6 0x x− − = . Also, find theequation whose roots are α + β, β + γand α + γ

To find the roots of the cubic equation 3760x^3 - 7x^2 - 6x = 0, we can use the factoring method or the quadratic formula. Let's use the factoring method.

Step 1: Factor out the common factor, if any. In this case, there is no common factor among the terms.

Step 2: Look for any possible rational roots. To find the possible rational roots, we can use the rational root theorem. According to the theorem, the possible rational roots are the factors of the constant term (in this case, 0) divided by the factors of the leading coefficient (in this case, 3760).

The factors of 0 are 0 and all real numbers. The factors of 3760 are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±40, ±47, ±80, ±94, ±188, ±235, ±376, ±470, ±752, ±940, ±1880, ±2350, ±3760.

Step 3: Test the possible rational roots. We substitute each possible rational root into the equation to check if it satisfies the equation.

By testing the possible rational roots, we find that x = 0 is a root of the equation.

Step 4: Divide the equation by the root found. We divide the equation by (x - 0) to obtain a quadratic equation.

After dividing, we get 3760x^2 - 6 = 0.

Step 5: Solve the quadratic equation. To solve the quadratic equation 3760x^2 - 6 = 0, we can use the quadratic formula.

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.

In this case, a = 3760, b = 0, and c = -6.

Substituting the values into the quadratic formula, we get x = ±√(6/3760).

Simplifying further, we get x = ±√(3/1880).

Therefore, the roots of the cubic equation 3760x^3 - 7x^2 - 6x = 0 are x = 0, x = √(3/1880), and x = -√(3/1880).

To find the equation whose roots are α + β, β + γ, and α + γ, we can use the sum and product of roots formulas.

The sum of roots formula states that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots α and β is given by α + β = -b/a.

In this case, the roots are α = 0, β = √(3/1880), and γ = -√(3/1880).

Using the sum of roots formula, we can find the equation whose roots are α + β, β + γ, and α + γ.

α + β = 0 + √(3/1880) = √(3/1880) β + γ = √(3/1880) + (-√(3/1880)) = 0 α + γ = 0 + (-√(3/1880)) = -√(3/1880)

Therefore, the equation whose roots are α + β, β + γ, and α + γ is x^2 - (√(3/1880))x - (√(3/1880)) = 0.