The quadratic inequality shows us in which interval the function is positive and in which it is negative - according to the inequality symbol. To solve quadratic inequalities correctly, it is convenient to remember two things:

The quadratic inequality shows us in which interval the function is positive and in which it is negative - according to the inequality symbol. To solve quadratic inequalities correctly, it is convenient to remember two things:

**Set of positivity and negativity of the function:**

Set of positivity - represents the $X$s in which the graph of the parabola is above the $X$ axis, with $Y$ value positive.

Set of negativity - represents the $X$s in which the graph of the parabola is below the $X$ axis, with $Y$ value negative.**Dividing by a negative term - reverses the sign of the inequality.**

- We will carry out the transposition of members and isolate the quadratic equation until one side equals 0. Remember that when we divide by a negative term, the inequality is reversed.
- Let's draw a diagram of the parabola - placing points of intersection with the $X$ axis and identifying the maximum and minimum of the parabola.
- Let's calculate the corresponding interval according to the exercise and the diagram.

Quadratic equation $>0βΆ$ Set of positivity

Quadratic equation $<0βΆ$ Set of negativity

Solve the following equation:

\( x^2+4>0 \)

$3x(x+1)<2(x^2+15)+2x$

**Solution:**

We will progress step by step:

1) Let's transpose terms and isolate the quadratic equation until one side equals $0$. Remember that when we divide by a negative term, the inequality is reversed.

In the first step, we will leave $0$ on one side of the equation.

Note that, in this exercise, we must first solve what appears between parentheses.

We will open the parentheses and obtain:

$3x^2+3x>2x^2+30+2x$

Now let's transpose terms and we will obtain:

$X^2+X-30>0$

Magnificent. We have left $0$ on one side. Let's continue to the second step.

2) Let's draw a diagram of the parabola - placing intersection points with the $X$ axis and identifying the maximum and minimum of the parabola.

Let's find the intersection points of the function with the $X$ axis:

According to the quadratic formula we will obtain:

$βββββββX=5,-6$

We will see that the extremity of the function is the minimum (smile) since the coefficient of $X^2$ is positive.

Let's draw a diagram:

3) Let's calculate the corresponding interval according to the exercise and the diagram. Β

In the exercise, we arrived at the following equation:

$βββββββX^2+X-30>0$

That is, we are looking for the intervals in which the function is greater than $0$. Its set of positivity.

We will ask ourselves: In which intervals is the function positive? At which $X$s is the graph of the function above the $X$ axis?

The answer is when

$X>5$

$X<-6$

**And these are the solutions for the quadratic inequality.**

**If you are interested in this article, you might also be interested in the following articles:**

Family of parabolas y=xΒ²+c: Vertical shift

Family of parabolas y=(x-p)Β²+k (combination of horizontal and vertical shift)

Vertex form of the quadratic function

Factored form of the quadratic function

Completing the square in a quadratic equation

Standard form of the quadratic function

System of quadratic equations - Algebraic and graphical solution

Solution of a system of equations when one is linear and the other quadratic

**In the** **Tutorela** **blog, you will find a variety of articles on mathematics.**

Solve the following equation:

x^2+4>0

All values of $x$

Solve the following equation:

x^2+9>0

All values of $x$

Solve the following equation:

-x^2-9>0

There is no solution.

Solve the following equation:

-x^2+2x>0

0 < x < 2

Solve the following equation:

x^2-3x+4<0

There is no solution.

Test your knowledge

Question 1

Solve the following equation:

\( x^2+9>0 \)

Question 2

Solve the following equation:

\( -x^2-9>0 \)

Question 3

Solve the following equation:

\( -x^2+2x>0 \)

Related Subjects

- Inequalities
- Inequalities with Absolute Value
- Absolute Value
- Absolute Value Inequalities
- Quadratice Equations and Systems of Quadraric Equations
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Word Problems
- The quadratic function
- Parabola
- Symmetry in a parabola
- Plotting the Quadratic Function Using Parameters a, b and c
- Finding the Zeros of a Parabola
- Methods for Solving a Quadratic Function
- Squared Trinomial
- The quadratic equation