7*x^2-40x>-60 inequation

Given the inequality:
$$7 x^{2} - 40 x > -60$$
To solve this inequality, we must first solve the corresponding equation:
$$7 x^{2} - 40 x = -60$$
Solve:
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$7 x^{2} - 40 x = -60$$
to
$$\left(7 x^{2} - 40 x\right) + 60 = 0$$
This equation is of the form

a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 7$$
$$b = -40$$
$$c = 60$$
, then

D = b^2 - 4 * a * c = 

(-40)^2 - 4 * (7) * (60) = -80

Because D<0, then the equation
has no real roots,
but complex roots is exists.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{20}{7} + \frac{2 \sqrt{5} i}{7}$$
Simplify
$$x_{2} = \frac{20}{7} - \frac{2 \sqrt{5} i}{7}$$
Simplify
$$x_{1} = \frac{20}{7} + \frac{2 \sqrt{5} i}{7}$$
$$x_{2} = \frac{20}{7} - \frac{2 \sqrt{5} i}{7}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$7 \cdot 0^{2} - 40 \cdot 0 > -60$$

0 > -60

so the inequality is always executed