# Solve 3x^2-9x+5<0.

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The equation y = 3x^2-9x+5 is that of a parabola.

As can be seen in the graph provided above, the parabola opens upwards and y is negative only for a certain set of values of x.

To determine the values for which y = 3x^2-9x+5 is less than 0, determine the roots of the equation 3x^2-9x+5 = 0. The values of x lying between the roots give a negative value for y = 3x^2-9x+5.

3x^2-9x+5 = 0

The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula `(-b+-sqrt(b^2 - 4ac))/(2a)`

Here, a = 3, b = -9 and c = 5, the of the equation are:

`(9+-sqrt(81 - 4*3*5))/(6)`

= `(9+-sqrt(21))/(6)`

= `(9+sqrt(21))/(6)` and `(9-sqrt(21))/(6)`

The set of values of x for which 3x^2-9x+5 < 0 is `((9+sqrt(21))/(6),(9-sqrt(21))/(6))`

To solve 3x^2-9x+5 <0.

Since the right side zero, the expression 3x^2-9x+5 is negative.

We find out the roots of the equation by quadratic formula:

x2 = {-(-9)+sqrt(9^2-4*3*5)}/2*3 = {9+sqrt21}/6

x1 = {-(-9)+sqrt(9^2-4*3*5)}/2*3 = {9-sqrt21}/6

Therefore 3x^2-9x+5 = 3(x-x1)(x-x2) .

Therefore 3x^2-9x+5 < 0 implies (x-x1)(x-x2) < 0.

Therefore x should be between the values x1 and x2 .

Therefore x1 < x< x2 .

Therefore {9-sqrt21}/6 < x < {9+sqrt21}/6.

One way to solve the inequality is to draw a graph of the function

f(x) = 3x^2-9x+5

The area between the roots is the area where the graph goes below the x axis. This area represents the solution of the inequality.

First, we'll determine the roots of the function:

3x^2-9x+5 = 0

We'll apply the quadratic formula:

x1 = [9+sqrt(81 - 60)]/6

x1 = (9+sqrt21)/6

x2 = (9-sqrt21)/6

**The expresison is negative when x is located in the interval **

**((9-sqrt21)/6 ; (9+sqrt21)/6)**