Question

prove that the roots of the equation x2+(2a-1) x+a2=0 are real if a1/4
$\leqslant$

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Answer 1

Question;

Prove that the roots of the equation

x^2 + (2a-2)x + a^2 = 0 are real if

a ≤ 1/4.

Solution :

The given quadratic equation is;

x^2 + (2a-2)x + a^2 = 0.

Thus,

Determinant = B^2 - 4•A•C

=> D = (2a - 1)^2 - 4•1•a^2

=> D = 4a^2 - 4a + 1 - 4a^2

=> D = 1 - 4a.

Also,

We know that,

For real roots of a quadratic equation,

its determinant must be greater than or equal to zero.

ie;

=> D ≥ 0

=> 1 - 4a ≥ 0

=> 1 ≥ 4a

=> 4a ≤ 1

=> a ≤ 1/4

Hence proved.