Math  /  Algebra

QuestionIf α\alpha and β\beta are the roots of the quadratic gquation x29x+2=0x^{2}-9 x+2=0, find the quadratic equation ane α3\alpha^{3} and β3\beta^{3}

Studdy Solution

STEP 1

Assumptions
1. The given quadratic equation is x29x+2=0x^2 - 9x + 2 = 0.
2. α\alpha and β\beta are the roots of the given quadratic equation.
3. We need to find the quadratic equation whose roots are α3\alpha^3 and β3\beta^3.

STEP 2

Using Vieta's formulas, we know that for the quadratic equation x29x+2=0x^2 - 9x + 2 = 0:
α+β=9\alpha + \beta = 9
αβ=2\alpha \beta = 2

STEP 3

We need to find the sum α3+β3\alpha^3 + \beta^3. We can use the identity for the sum of cubes:
α3+β3=(α+β)(α2αβ+β2)\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)

STEP 4

We already know α+β=9\alpha + \beta = 9. Next, we need to find α2+β2\alpha^2 + \beta^2. Using the identity:
α2+β2=(α+β)22αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta

STEP 5

Substitute the known values into the identity:
α2+β2=9222\alpha^2 + \beta^2 = 9^2 - 2 \cdot 2

STEP 6

Calculate α2+β2\alpha^2 + \beta^2:
α2+β2=814=77\alpha^2 + \beta^2 = 81 - 4 = 77

STEP 7

Now substitute α+β\alpha + \beta, αβ\alpha \beta, and α2+β2\alpha^2 + \beta^2 into the sum of cubes identity:
α3+β3=9(772)\alpha^3 + \beta^3 = 9(77 - 2)

STEP 8

Simplify the expression:
α3+β3=975=675\alpha^3 + \beta^3 = 9 \cdot 75 = 675

STEP 9

Next, we need to find the product α3β3\alpha^3 \beta^3. Using the property of exponents:
α3β3=(αβ)3\alpha^3 \beta^3 = (\alpha \beta)^3

STEP 10

Substitute the known value of αβ\alpha \beta:
α3β3=23\alpha^3 \beta^3 = 2^3

STEP 11

Calculate α3β3\alpha^3 \beta^3:
α3β3=8\alpha^3 \beta^3 = 8

STEP 12

Now we have the sum and product of the roots of the new quadratic equation. The new quadratic equation with roots α3\alpha^3 and β3\beta^3 can be written as:
x2(α3+β3)x+α3β3=0x^2 - (\alpha^3 + \beta^3)x + \alpha^3 \beta^3 = 0

STEP 13

Substitute the calculated values into the quadratic equation:
x2675x+8=0x^2 - 675x + 8 = 0
The quadratic equation whose roots are α3\alpha^3 and β3\beta^3 is:
x2675x+8=0\boxed{x^2 - 675x + 8 = 0}

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