#### Question

Determine the nature of the roots of the following quadratic equation:

2(a^{2} + b^{2})x^{2} + 2(a + b)x + 1 = 0

#### Solution

The given equation is

2(a^{2} + b^{2})x^{2} + 2(a + b)x + 1 = 0

The given equation is in the form of ax^{2} + bx + c = 0

where a = 2(a^{2} + b^{2}), b = 2(a + b), c = 1

Therefore the discriminant

D = b^{2} - 4ac

= (2(a + b))^{2} - 4 x (2(a^{2} + b^{2})) x (1)

= 4(a + b)^{2} - 8a^{2} - 8b^{2}

= 4(a^{2} + b^{2} + 2ab) - 8a^{2} - 8b^{2}

= 4a^{2} + 4b^{2} + 8ab - 8a^{2} - 8b^{2}

= 8ab - 4a^{2} - 4b^{2}

∵ D < 0,

∴ The roots of the given equation are not real.

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#### APPEARS IN

Solution Determine the Nature of the Roots of the Following Quadratic Equation: 2(A2 + B2)X2 + 2(A + B)X + 1 = 0 Concept: Nature of Roots.