Detail solution

Given the inequality:

$$x^{2} - 29 x < 0$$

To solve this inequality, we must first solve the corresponding equation:

$$x^{2} - 29 x = 0$$

Solve:

This equation is of the form

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

$$a = 1$$

$$b = -29$$

$$c = 0$$

, then

Because D > 0, then the equation has two roots.

or

$$x_{1} = 29$$

$$x_{2} = 0$$

$$x_{1} = 29$$

$$x_{2} = 0$$

$$x_{1} = 29$$

$$x_{2} = 0$$

This roots

$$x_{2} = 0$$

$$x_{1} = 29$$

is the points with change the sign of the inequality expression.

First define with the sign to the leftmost point:

$$x_{0} < x_{2}$$

For example, let's take the point

$$x_{0} = x_{2} - \frac{1}{10}$$

=

$$- \frac{1}{10}$$

=

$$- \frac{1}{10}$$

substitute to the expression

$$x^{2} - 29 x < 0$$

$$\left(- \frac{1}{10}\right)^{2} - \frac{\left(-1\right) 29}{10} < 0$$

but

Then

$$x < 0$$

no execute

one of the solutions of our inequality is:

$$x > 0 \wedge x < 29$$

$$x^{2} - 29 x < 0$$

To solve this inequality, we must first solve the corresponding equation:

$$x^{2} - 29 x = 0$$

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

$$a = 1$$

$$b = -29$$

$$c = 0$$

, then

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

(-29)^2 - 4 * (1) * (0) = 841

Because D > 0, then the equation has two roots.

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

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

or

$$x_{1} = 29$$

$$x_{2} = 0$$

$$x_{1} = 29$$

$$x_{2} = 0$$

$$x_{1} = 29$$

$$x_{2} = 0$$

This roots

$$x_{2} = 0$$

$$x_{1} = 29$$

is the points with change the sign of the inequality expression.

First define with the sign to the leftmost point:

$$x_{0} < x_{2}$$

For example, let's take the point

$$x_{0} = x_{2} - \frac{1}{10}$$

=

$$- \frac{1}{10}$$

=

$$- \frac{1}{10}$$

substitute to the expression

$$x^{2} - 29 x < 0$$

$$\left(- \frac{1}{10}\right)^{2} - \frac{\left(-1\right) 29}{10} < 0$$

291 --- < 0 100

but

291 --- > 0 100

Then

$$x < 0$$

no execute

one of the solutions of our inequality is:

$$x > 0 \wedge x < 29$$

_____ / \ -------ο-------ο------- x2 x1

Solving inequality on a graph