Question

Find the nature of the roots of the following equation without finding the roots.

$$ 2x^2 - 8x + 3 = 0 $$

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Answer

To determine the nature of the roots of the quadratic equation, it is essential to calculate the discriminant without solving for the actual roots. Let's consider the given equation:

$$ 2x^2 - 8x + 3 = 0 $$

A quadratic equation can be generalized in the form:

$$ ax^2 + bx + c = 0 $$

By comparing this general equation with the given equation, we get:

  • ( a = 2 )
  • ( b = -8 )
  • ( c = 3 )

The discriminant (( D )) of a quadratic equation is given by:

$$ D = b^2 - 4ac $$

Substitute the values of ( a ), ( b ), and ( c ) into the discriminant formula:

$$ D = (-8)^2 - 4 \cdot 2 \cdot 3 $$

Let's calculate this step-by-step:

  1. Compute ( (-8)^2 ):

$$ (-8)^2 = 64 $$

  1. Compute ( 4 \cdot 2 \cdot 3 ):

$$ 4 \cdot 2 \cdot 3 = 24 $$

  1. Subtract the second result from the first:

$$ 64 - 24 = 40 $$

Thus, the discriminant ( D ) is:

$$ D = 40 $$

Since the discriminant ( D ) is greater than 0 (( 40 > 0 )), this indicates that the equation has:

  • Two real roots
  • The roots are distinct and unequal

Moreover, because the discriminant is not a perfect square (40 is not a perfect square), the roots will be irrational and will form a pair of conjugate surds.

Conclusion:

The nature of the roots of the quadratic equation ( 2x^2 - 8x + 3 = 0 ) are real, distinct, and irrational.


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