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What is the sum of the squares of the roots of the equation x2 + 2x - 143 = 0?
1. 170
2. 180
3. 190
4. 290
5. None of these

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Correct Answer - Option 4 : 290

Concept:

Let us consider the standard form of a quadratic equation, ax2 + bx + c = 0

Let α and β be the two roots of the above quadratic equation. 

  • The sum of the roots of a quadratic equation given by: \({\rm{\alpha }} + {\rm{\beta }} = - \frac{{\rm{b}}}{{\rm{a}}} = - \frac{{{\rm{coefficient\;of\;x}}}}{{{\rm{coefficient\;of\;}}{{\rm{x}}^2}}}\)
  • The product of the roots is given by: \({\rm{\alpha \beta }} = \frac{{\rm{c}}}{{\rm{a}}} = \frac{{{\rm{constant\;term}}}}{{{\rm{coefficient\;of\;}}{{\rm{x}}^2}}}\)
 

Calculation:

Given equation is x2 + 2x - 143 = 0

To Find: Sum of the squares of the roots

Let α and β be the two roots of the above quadratic equation.

Now, Sum of roots = α + β = -2

Product of roots = αβ = -143

(α + β)2 = α2 + β2 + 2αβ

⇒ (-2)2 = α2 + β2 + 2 × (-143)

⇒ 4 = α2 + β2 - 286

∴ Sum of the squares of the roots = α2 + β2 = 290 

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