To determine the value of b in the second and third expressions, we can set them equal to zero and then solve for x.
For the second expression:
5x^2 + bx + 20 = 0
This is a quadratic equation that can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
where a = 5, b = b, and c = 20.
The discriminant (b^2 - 4ac) must be greater than or equal to zero for the equation to have real roots. So,
b^2 - 4ac ≥ 0b^2 - 4(5)(20) ≥ 0b^2 - 80 ≥ 0b^2 ≥ 80b ≥ √80b ≥ 8.94 (approximately)
Therefore, b must be greater than or equal to 8.94.
For the third expression:
3x^2 + bx + 16 = 0
Similarly, we have:
b^2 - 4ac ≥ 0b^2 - 4(3)(16) ≥ 0b^2 - 192 ≥ 0b^2 ≥ 192b ≥ √192b ≥ 13.85 (approximately)
Therefore, for the third expression, b must be greater than or equal to 13.85.
To determine the value of b in the second and third expressions, we can set them equal to zero and then solve for x.
For the second expression:
5x^2 + bx + 20 = 0
This is a quadratic equation that can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
where a = 5, b = b, and c = 20.
The discriminant (b^2 - 4ac) must be greater than or equal to zero for the equation to have real roots. So,
b^2 - 4ac ≥ 0
b^2 - 4(5)(20) ≥ 0
b^2 - 80 ≥ 0
b^2 ≥ 80
b ≥ √80
b ≥ 8.94 (approximately)
Therefore, b must be greater than or equal to 8.94.
For the third expression:
3x^2 + bx + 16 = 0
Similarly, we have:
b^2 - 4ac ≥ 0
b^2 - 4(3)(16) ≥ 0
b^2 - 192 ≥ 0
b^2 ≥ 192
b ≥ √192
b ≥ 13.85 (approximately)
Therefore, for the third expression, b must be greater than or equal to 13.85.