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:
Solve the following equation:
\( x^2+4>0 \)
Solution:
We will progress step by step:
1) Let's transpose terms and isolate the quadratic equation until one side equals . Remember that when we divide by a negative term, the inequality is reversed.
In the first step, we will leave 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:
Now let's transpose terms and we will obtain:
Magnificent. We have left on one side. Let's continue to the second step.
2) Let's draw a diagram of the parabola - placing intersection points with the axis and identifying the maximum and minimum of the parabola.
Let's find the intersection points of the function with the axis:
According to the quadratic formula we will obtain:
We will see that the extremity of the function is the minimum (smile) since the coefficient of is positive.
Let's draw a diagram:
3) Let's calculate the corresponding interval according to the exercise and the diagram.
In the exercise, we arrived at the following equation:
That is, we are looking for the intervals in which the function is greater than . Its set of positivity.
We will ask ourselves: In which intervals is the function positive? At which s is the graph of the function above the axis?
The answer is when
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=x²+c: Vertical shift
Family of parabolas y=(x-p)²+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
To solve this problem, let's examine the inequality .
The expression consists of two terms: and . Notice that:
Combining these observations, we see that:
Thus, there are no values of for which the expression is zero or negative. Instead, the expression is always positive for all real numbers .
Therefore, the solution to the inequality is all values of .
All values of
Solve the following equation:
x^2+9>0
Let's explore this problem step-by-step:
The inequality given is .
1. To understand this inequality, we start by considering the expression . We know that for any real number , . This means is always non-negative.
2. Since for every real number, adding 9 to will necessarily make the expression greater than zero, because a non-negative number plus a positive number gives a positive result: .
3. Therefore, the inequality holds true for all real numbers . There is no value of that makes the left side equal to or less than zero.
4. Thus, the solution to the inequality is that it holds for all values of .
Consequently, the correct choice from the options provided is:
Therefore, the solution is that the inequality is true for all values of .
All values of
Solve the following equation:
-x^2+2x>0
To solve the inequality , we begin by considering the corresponding equation .
First, factor the quadratic equation:
These roots divide the number line into three intervals: , , and .
We need to test these intervals to determine where the inequality holds:
Thus, the inequality is satisfied for the interval .
Therefore, the solution to the inequality is , which corresponds to choice 2 in the given options.
0 < x < 2
Solve the following equation:
-x^2-9>0
To solve this quadratic inequality, , we will follow these steps:
Let's analyze the equation:
Rewrite the inequality:
Add 9 to both sides:
Multiply the entire inequality by and remember to reverse the inequality sign:
Observe the inequality :
Note that , being a square of any real number, is always greater than or equal to zero.
As cannot be less than negative nine for any real number , the inequality has no solution in the realm of real numbers.
Therefore, the correct answer is:
There is no solution.
There is no solution.
Solve the following equation:
x^2-8x+12>0
Let's proceed to solve the inequality .
The factorization gives us the critical points and . These points divide the number line into three intervals: , , and .
Now, we evaluate the sign of the product in each interval:
The inequality holds for and .
Thus, the solution to the inequality is or .
Therefore, the correct answer is .
x < 2,6 < x
Solve the following equation:
\( -x^2+2x>0 \)
Solve the following equation:
\( -x^2-9>0 \)
Solve the following equation:
\( x^2+9>0 \)