If alpha, beta are the roots of 6x^2 - 6x + 1 = 0, then 1/2(a + b*alpha + c*alpha^2 + d*alpha^3) + 1/2(a + b*beta + c*beta^2 + d*beta^3) =

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q10 – AP-EAMCET 2026

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Home Maths Complex & QuadraticQuestion #54

α, β

are the roots of

6 x^{2} - 6 x + 1 = 0

, then

\frac{1}{2} (a + b α + c α^{2} + d α^{3}) + \frac{1}{2} (a + b β + c β^{2} + d β^{3}) =

Clue 1 of 3: Sum and Product of Roots

For a quadratic equation $A x^{2} + B x + C = 0$ , the sum of roots is $- (B / A)$ and the product of roots is $C / A$ .

🧠 Full Solution Path

Step 1: Identify Sum and Product of Roots✦ Active

For the quadratic equation $6 x^{2} - 6 x + 1 = 0$ , we can find the sum and product of its roots, $α$ and $β$ .

α + β = - \frac{(- 6)}{6} = 1

α β = \frac{1}{6}

Step 2: Simplify the Given Expression○ Expand

Combine the two terms in the given expression by factoring out $\frac{1}{2}$ and grouping terms with common coefficients $a, b, c, d$ .

E = \frac{1}{2} (a + b α + c α^{2} + d α^{3} + a + b β + c β^{2} + d β^{3})

E = \frac{1}{2} (2 a + b (α + β) + c (α^{2} + β^{2}) + d (α^{3} + β^{3}))

💡 Teacher's Secret Hint

Remember that the expression is symmetric with respect to $α$ and $β$ , which often simplifies calculations.

Step 3: Calculate Higher Powers of Roots○ Expand

Now, we need to calculate $α^{2} + β^{2}$ and $α^{3} + β^{3}$ using the sum and product of roots from Step 1.

α^{2} + β^{2} = (α + β)^{2} - 2 α β = (1)^{2} - 2 (\frac{1}{6}) = 1 - \frac{1}{3} = \frac{2}{3}

α^{3} + β^{3} = (α + β) (α^{2} - α β + β^{2}) = (α + β) [(α^{2} + β^{2}) - α β]

= (1) [(\frac{2}{3}) - (\frac{1}{6})] = \frac{2}{3} - \frac{1}{6} = \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}

💡 Teacher's Secret Hint

Make sure to correctly apply the algebraic identities. A common mistake is using $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$ , which also works and might be quicker.

Step 4: Substitute and Final Calculation○ Expand

Substitute the calculated values back into the simplified expression from Step 2.

E = \frac{1}{2} (2 a + b (1) + c (\frac{2}{3}) + d (\frac{1}{2}))

E = a + \frac{b}{2} + \frac{c}{3} + \frac{d}{4}

💡 Teacher's Secret Hint

Double-check your arithmetic, especially when dealing with fractions. Ensure all terms are correctly multiplied by $\frac{1}{2}$ .

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Maths Question 10 – AP-EAMCET 2026

Clue 1 of 3: Sum and Product of Roots

🧠 Full Solution Path