the least positive value of k for which the quadratic equation 2x^2 kx-4=0
What is the least positive value of ( k ) for which the quadratic equation ( 2x^2 + kx - 4 = 0 ) has real roots?
Answer:
To find the least positive value of ( k ) such that the quadratic equation ( 2x^2 + kx - 4 = 0 ) has real roots, we need to ensure that the discriminant of the quadratic equation is non-negative.
Understanding the Discriminant
For a quadratic equation in the form ( ax^2 + bx + c = 0 ), the discriminant ( \Delta ) is given by:
[
\Delta = b^2 - 4ac
]
The roots of the equation are real if ( \Delta \geq 0 ).
Applying to Our Equation
Given:
- ( a = 2 )
- ( b = k )
- ( c = -4 )
Substitute these values into the discriminant formula:
[
\Delta = k^2 - 4 \times 2 \times (-4)
]
Simplify:
[
\Delta = k^2 + 32
]
Condition for Real Roots
The equation will have real roots when:
[
k^2 + 32 \geq 0
]
This inequality is always true since ( k^2 ) is always non-negative and ( 32 ) is positive. The discriminant will never be negative, so any real number ( k ) will make the quadratic have real roots.
However, if looking for the least positive value of ( k ), you just need ( k > 0 ).
Since ( k ) must be positive, the smallest value ( k ) can take is any positive value, typically, the smallest strictly positive number greater than zero.
Summary: Any positive value of ( k ) will work since the quadratic will always have real roots due to the non-negative discriminant derived. Therefore, the smallest positive value can be any value slightly greater than zero like ( k = 0.01 ) or something similar.