7х^2-10х+7>0
To solve the quadratic inequality \(7x^2 - 10x + 7 > 0\), we first need to find the roots of the corresponding quadratic equation \(7x^2 - 10x + 7 = 0\).
We will use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 7\), \(b = -10\), and \(c = 7\).
- Calculate the discriminant \(D\): \[ D = b^2 - 4ac = (-10)^2 - 4(7)(7) = 100 - 196 = -96 \]
Since the discriminant is negative (\(D < 0\)), the quadratic equation has no real roots. Therefore, the expression \(7x^2 - 10x + 7\) does not intersect the x-axis.
Determine the sign of the quadratic expression: Since the coefficient of \(x^2\) (which is \(7\)) is positive, the quadratic opens upwards. Because it has no real roots, it will be positive for all \(x\).
Conclusion: The inequality \(7x^2 - 10x + 7 > 0\) holds true for all real numbers. Thus, the solution is: \[ \boxed{(-\infty, \infty)} \]
