In this page we are going to discuss about quadratic inequalities concept.A quadratic inequality in one variable is in the form of an expression:
ax2 + bx + c ≤ 0 or ax2 + bx + c < 0
ax2 + bx + c ≥ 0 or ax2 + bx + c < 0
where a, b and c are real numbers, a ≠ 0. The values of x which satisfy the given inequality are called the solutions of the inequality.
A quadratic equation has only two roots. But a quadratic inequality has many roots.
Methods to solve quadratic Inequalities
There are two methods to solve quadratic inequalities -
Method 1: Finding the solution by dividing the given polynomial into factors. This method is called the 'Algebraic method'.
Method 2: Finding the solutions by drawing the graph of the inequality. This method is called the 'Graphical method'.
Note:
Solving quadratic inequalities
Below you can see the example on solving quadratic inequalities-
Example:1 Find the solution set of x2 – 4x – 21 ≥ 0 in
1. Algebraic Method
2. Graphical Method
Solution:
1. Algebraic Method:
X2 - 4x - 21 ≥ 0
=> ( x + 3) ( x - 7 ) ≥ 0
Here the coefficient of x2 is > 0 and the sign of the quadratic expression is non-negative.
So, x ≥ -3 and x ≤ 7 or there are two cases for the product ( x + 3 ) ( x - 7 ) to be non-negative.
Case (1):
x + 3 ≥ 0 and x - 7 ≥ 0
=> x ≥ -3 and x ≥ 7
So it is necessary that x≥ 7. Notice the line marked below with x ≥ -3 and x ≥ 7 and their intersection.
From the figure above, it is clear that the intersection is x ≥ 7.
Case (2):
x + 3 ≤ 0 and x - 7 ≤ 0
=> x ≤ -3 and x ≤ 7
So it is clear x ≤ -3.
i.e., the values of x for which x2 - 4x - 21 ≥ 0 are given by these two cases.
All values of x satisfying the in equations x ≤ -3 or x ≥ 7 become the solution set for the given in equations. This solution set can be shown in the graph given below.
From the figure above, it is clear that the intersection is x ≥ 7.
2. Graphical Method:
The given polynomial function is f(x) = x2 - 4x - 21
Using the above chart, we can draw a graph given below
Now, we require the values of x satisfying x2 - 4x - 21 ≥ 0 , i.e., f(x) ≥ 0 ( y ≥ 0 ). That is we require the values of x for which f(x) lies on x-axis and above. But it is clearly understood from the figure that the values of x are -3, left side to -3, +7, and right side to +7. Thus all values of x such that x ≤ 3 and x ≥ 7 become the solution set.
ax2 + bx + c ≤ 0 or ax2 + bx + c < 0
ax2 + bx + c ≥ 0 or ax2 + bx + c < 0
where a, b and c are real numbers, a ≠ 0. The values of x which satisfy the given inequality are called the solutions of the inequality.
A quadratic equation has only two roots. But a quadratic inequality has many roots.
Methods to solve quadratic Inequalities
There are two methods to solve quadratic inequalities -
Method 1: Finding the solution by dividing the given polynomial into factors. This method is called the 'Algebraic method'.
Method 2: Finding the solutions by drawing the graph of the inequality. This method is called the 'Graphical method'.
Note:
- The trick in solving a quadratic inequality is to replace the inequality symbol with an equal sign and solve the resulting equation. The solutions to the equation will allow us to establish intervals that will let you solve the inequality.
- Plot the solutions on number line creating the intervals for investigation. Pick any number from each interval and test it in original inequality. If the result is true, that interval is the solution to the inequality.
Solving quadratic inequalities
Below you can see the example on solving quadratic inequalities-
Example:1 Find the solution set of x2 – 4x – 21 ≥ 0 in
1. Algebraic Method
2. Graphical Method
Solution:
1. Algebraic Method:
X2 - 4x - 21 ≥ 0
=> ( x + 3) ( x - 7 ) ≥ 0
Here the coefficient of x2 is > 0 and the sign of the quadratic expression is non-negative.
So, x ≥ -3 and x ≤ 7 or there are two cases for the product ( x + 3 ) ( x - 7 ) to be non-negative.
Case (1):
x + 3 ≥ 0 and x - 7 ≥ 0
=> x ≥ -3 and x ≥ 7
So it is necessary that x≥ 7. Notice the line marked below with x ≥ -3 and x ≥ 7 and their intersection.
From the figure above, it is clear that the intersection is x ≥ 7.
Case (2):
x + 3 ≤ 0 and x - 7 ≤ 0
=> x ≤ -3 and x ≤ 7
So it is clear x ≤ -3.
i.e., the values of x for which x2 - 4x - 21 ≥ 0 are given by these two cases.
All values of x satisfying the in equations x ≤ -3 or x ≥ 7 become the solution set for the given in equations. This solution set can be shown in the graph given below.
From the figure above, it is clear that the intersection is x ≥ 7.
2. Graphical Method:
The given polynomial function is f(x) = x2 - 4x - 21
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | … |
| f(x) | 0 | -9 | -16 | -21 | -24 | -25 | -24 | -21 | … |
Using the above chart, we can draw a graph given below
Now, we require the values of x satisfying x2 - 4x - 21 ≥ 0 , i.e., f(x) ≥ 0 ( y ≥ 0 ). That is we require the values of x for which f(x) lies on x-axis and above. But it is clearly understood from the figure that the values of x are -3, left side to -3, +7, and right side to +7. Thus all values of x such that x ≤ 3 and x ≥ 7 become the solution set.
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