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 x^{2} + 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} = α^{2} + β^{2} + 2αβ

⇒ (-2)^{2} = α^{2} + β^{2} + 2 × (-143)

⇒ 4 = α^{2} + β^{2} - 286

∴ Sum of the squares of the roots = α^{2} + β^{2} = 290