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.
Question
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.
Solution
To find the roots of the cubic equation 3760x^3 - 7x^2 - 6x = 0, we can use the factoring method or the quadratic formula.
Step 1: Factor out the common factor, if possible. In this case, there is no common factor among the terms.
Step 2: Use the factoring method or the quadratic formula. Since the equation is a cubic equation, we can use the factoring method or the quadratic formula to find the roots.
Factoring method: To factor the equation, we need to find the factors of the constant term (0) that add up to the coefficient of the quadratic term (-7).
The factors of 0 are 0 and any real number.
Therefore, one of the roots is x = 0.
Quadratic formula: The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
In our equation, a = 3760, b = -7, and c = -6.
Using the quadratic formula, we can find the other two roots:
x = (-(-7) ± √((-7)^2 - 4(3760)(-6))) / (2(3760)) x = (7 ± √(49 + 90480)) / 7520 x = (7 ± √90489) / 7520
Simplifying further, we have: x = (7 ± 301) / 7520
So, the other two roots are: x = (7 + 301) / 7520 = 308 / 7520 = 0.0409 x = (7 - 301) / 7520 = -294 / 7520 = -0.039
Therefore, the roots of the cubic equation 3760x^3 - 7x^2 - 6x = 0 are: α = 0 β = 0.0409 γ = -0.039
To find the equation whose roots are α + β, β + γ, and α + γ, we can use the sum and product of roots formula.
The sum of roots formula states that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is given by -b/a.
In our case, the sum of the roots α + β, β + γ, and α + γ is: (α + β) + (β + γ) + (α + γ) = 2(α + β + γ)
So, the equation whose roots are α + β, β + γ, and α + γ is: 2(α + β + γ) = 0
Simplifying further, we have: 2(0 + 0.0409 - 0.039) = 0 2(0.0019) = 0 0.0038 = 0
Therefore, the equation whose roots are α + β, β + γ, and α + γ is 0.
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