Find the nature of the roots of the following equation without finding the roots. 2x^2 - 8x + 3 = 0
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:
- Compute ( (-8)^2 ):
$$ (-8)^2 = 64 $$
- Compute ( 4 \cdot 2 \cdot 3 ):
$$ 4 \cdot 2 \cdot 3 = 24 $$
- 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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