The standard form of a quadratic equation:
ax^{2} + bx + c = 0
Now let’s replace the equal sign with the inequality sign. Then we get the quadratic inequality in standard form.
An inequality with a quadratic equation expression is a quadratic inequality. You already know the standard form of a quadratic inequality. The graph of a quadratic function looks like this:
f (x )=ax^{2}+bx+c=0
And this is known as a parabola.
That means, when the quadratic equation is ax^{2} + bx + c > 0, it means f (x) > 0. In other words, the parabola is above the y-axis.
Here is the graphical representation of the parabola above the y-axis:
The graphs of quadratic inequalities follow the same relationship as graphs and equations of linear inequalities. Here are some graphical representations of quadratic inequalities:
Let’s consider the quadratic inequality y > x^{2} -1
As the second image of Figure 2 shows, the yellow region shows the graph of the quadratic inequality. The line segment from (-1,2) to (1,2) shows the solution itself graphically.
Now consider the third graph from Figure 2. It follows the same quadratic equation only with the < sign.
Remember, the graph of the inequality always overlaps with the x-axis.
Sounds confusing? Here is a table to make the formulas for quadratic inequality easier for you to understand.
Greater than inequality |
Less than inequality |
0 > ax² + bx + c |
0 < ax² + bx + c |
Solution: {x|r1 < X < r2} |
Solution: {x| x < r1 or x > r2} |
By now, you must have a clear idea about what quadratic inequalities are and the general formulas involved while graphing the same. So, now let’s have a look at some examples of how to graph a quadratic inequality.
Step 1: Assume it is a normal quadratic equation and solve this quadratic inequality.
So, the equation would look something like this:
0 = x^{2} + 2x - 3
0 = (x -1)(x + 3)
X = 1; x= -3
Step 2: Now graph the inequality on a number line.
Where r1 is -3 and r2 is 1.
Step 3: Write down the solution set.
{x| -3 < X < 1}
Step 1: Write the inequality with zero on one side in standard form.
Step 2: Graph the quadratic equation.
Step 3: Shade the regions on the graph that give you the desired results. (where the equation is greater and not equal to zero)
Step 4: Convert the shading to interval notation.
Notations Used In Quadratic Inequalities
The common notations used in quadratic inequalities are:
Symbol |
Words |
Example |
> |
Greater than |
X^{2} + 3x >2 |
< |
Less than |
7x^{2} < 28 |
≥ |
Greater than or equal to |
5 ≥ x^{2} - x |
≤ |
Less than or equal to |
2y^{2} + 1 ≤ 7y |
Example: Graphically solve x^{2} – 6x + 8 < 0. Write the solution in interval notation.
x^{2}−6x+8<0
f(x) = x^{2}−6x+8
a = 1, b = -6, c = 8
The parabola opens upward since a is positive.
Find out the axis of symmetry
x = -b/2a
x = 3.
f(3)=(3)^{2}−6(3)+8
f(3)= -1
The vertex hence is (3, -1)
f(0)-8
Hence, the y-intercept is (0,8)
The point is (6, 8).
Now if we solve the quadratic equation by factoring, we get:
f(x) = x2 – 6x +8
x = 2 or x = 4.
The x-intercepts are (2, 0) and (4, 0).
x^{2}−6x+8 < 0
The solution is (2, 4) in interval notation.
That’s all! This is the basic concept of graphing quadratic inequality. Start practising the problems once you get a solid grip on the formulas.
Ans.You can solve quadratic inequalities graphically or algebraically. The latter is easy and requires the right use of formula. The former method, however, involves the following steps:
Ans.The steps for solving quadratic equations by factorization method are:
Consider the equation- y ≤ x^{2}−x−12
The related equation is y = x^{2}−x−12
Factoring the right side we get y = (x +3)(x - 4)
The x-intercepts are at -3 and 4. The x coordinate of the vertex is 0.5.
y = (0.5 + 3)(0.5 -4)
y = (3.5)(-3.5)
y = -12.25
The vertex is at (0.5, -12.25)
Ans.The first step in solving quadratic inequality, whether graphically or algebraically, is to:
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