If α and β are the roots of the quadratic equation 3x² − 4x − 2 = 0, find the value of 4α + 4β / α − β

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The given quadratic equation is 3x² − 4x − 2 = 0. For a quadratic equation in the form ax² + bx + c = 0, the sum of the roots (α + β) and the product of the roots (α β) are given by Vieta's formulas: α + β = −b/a; α β = c/a; From the given equation, we have a=3, b=−4, and c=−2. Step 1: Calculate the sum and product of the roots. Sum of roots: α + β = −(−4)/(3) = 4/3 Product of roots: α β = (−2)/(3) i) Find the value of 4α + 4β Step 2: Factor out 4 and substitute the sum of roots. 4α + 4β = 4(α + β); 4(α + β) = 4 × 4/3; 4(α + β) = 16/3 The value is 16/3. ii) Find the value of α − β Step 3: Use the identity (α − β)² = (α + β)² − 4αβ. Substitute the values of α + β and α β: (α − β)² = (4/3)² − 4((−2)/(3)); (α − β)² = 16/9 + 8/3 To add the fractions, find a common denominator, which is 9: (α − β)² = 16/9 + (8 × 3)/(3 × 3); (α − β)² = 16/9 + 24/9; (α − β)² = (16 + 24)/(9); (α − β)² = 40/9 Step 4: Take the square root to find α − β. α − β = ± √(40/9); α − β = ± (√(40))/(√(9)); α − β = ± (√(4 × 10))/(3); α − β = ± (2√(10))/(3) The value is ± (2√(10))/(3).

If α and β are the roots of the quadratic equation 3x² − 4x − 2 = 0, find the value of 4α + 4β / α - β

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Step-by-step solution

Part i· 1 step

Factor out 4 and substitute the sum of roots.

Part ii· 2 steps
$\pm \frac{2\sqrt{10}}{3}$

Use the identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$ .

Take the square root to find $\alpha - \beta$ .

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