Mister Exam

# Square Inequality Step-by-Step

For example, you have entered (calculator here):
$$\left(25 x^{2} - 30 x\right) + 9 > 0$$

### Rapid solution

$$x > -\infty \wedge x < \infty \wedge x \neq \frac{3}{5}$$ $$x\ in\ \left(-\infty, \frac{3}{5}\right) \cup \left(\frac{3}{5}, \infty\right)$$

### Detail solution

Given the inequality:
$$\left(25 x^{2} - 30 x\right) + 9 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(25 x^{2} - 30 x\right) + 9 = 0$$
Solve:
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
$$a = 25$$
$$b = -30$$
$$c = 9$$
, then
D = b^2 - 4 * a * c =

(-30)^2 - 4 * (25) * (9) = 0

Because D = 0, then the equation has one root.
x = -b/2a = --30/2/(25)

$$x_{1} = \frac{3}{5}$$
$$x_{1} = \frac{3}{5}$$
$$x_{1} = \frac{3}{5}$$
This roots
$$x_{1} = \frac{3}{5}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{5}$$
=
$$\frac{1}{2}$$
substitute to the expression
$$\left(25 x^{2} - 30 x\right) + 9 > 0$$
$$\left(- \frac{30}{2} + 25 \left(\frac{1}{2}\right)^{2}\right) + 9 > 0$$
1/4 > 0

the solution of our inequality is:
$$x < \frac{3}{5}$$
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