Question

При каких значениях m неравенство mx^2-9mx+5m+1>0 выполняется при всех действительных значениях x

Answer 1

Если дискриминант уравнения меньше  0, то уравнение всегда будет больше 0 при любом значении х. Найдем дискриминант
m*x²-9mx+5m=0
D=(9m)²-4*m*5m=81m²-20m²=71m²
71m²<0
m²<0
m не имеет значений

Answer 2

Quadratic Inequality Analysis

To determine the values of \( m \) for which the inequality \( mx^2 - 9mx + 5m + 1 > 0 \) holds for all real values of \( x \), we can use the discriminant of the quadratic equation.

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula: \[ D = b^2 - 4ac \]

For the given inequality \( mx^2 - 9mx + 5m + 1 > 0 \), the discriminant is: \[ D = (-9m)^2 - 4m(5m + 1) \]

Conditions for the Inequality to Hold

For the inequality \( mx^2 - 9mx + 5m + 1 > 0 \) to hold for all real values of \( x \), the following conditions must be met:

1. The discriminant \( D \) must be less than zero, as this indicates that the quadratic equation has no real roots, and hence, the inequality holds for all real values of \( x \). 2. The coefficient of \( x^2 \) (which is \( m \)) must be positive, as this ensures that the parabola opens upwards, leading to the inequality holding for all real values of \( x \).

Conclusion

Therefore, the values of \( m \) for which the inequality \( mx^2 - 9mx + 5m + 1 > 0 \) holds for all real values of \( x \) are when the discriminant \( D \) is less than zero and the coefficient of \( x^2 \) (which is \( m \)) is positive.

Note: The specific values of \( m \) that satisfy these conditions can be calculated by solving the inequality and the discriminant condition simultaneously.