SOLVING QUADRATIC  INEQUALITIES GRAPHICALLY

About "Solving quadratic inequalities graphically"

"Solving quadratic inequalities graphically" is sometimes a difficult task for some students who study high school math.

Let us see the steps involved in solving quadratic inequality graphically.

Step 1 :

Let the given quadratic inequality be ax² + bx + c ≥ 0.

We have to write the quadratic function y = ax² + bx + c.

Step 2 :

The graph of  y = ax² + bx + c will either be open upward or downward parabola.

We can use the below table to know whether it is open upward or down ward. 

Step 3 :

The graph of  y = ax² + bx + c may intersect X- axis or not.

We can use the below table to know whether it intersects X -axis or not.

Step 4 :

We can use the formula x = -b/2a and find the vertex of the parabola. 

Step 5 :

We can use the results of step 2, step 3, step 4 and sketch the graph of the given parabola.

Step 6 :

We can get solution for the given inequality from the graph of the parabola.

Examples

To have better understanding on the above mentioned stuff, let us look at some example problems on "Solving quadratic inequalities graphically" 

Example 1 :

Solve the quadratic inequality given below graphically.

x² + 5x + 6 ≥ 0

Solution :

Step 1 :

Let y = x² + 5x + 6.

Then we have, x² + 5x + 6 ≥ 0 --------> y 0

Step 2 :

The sign of  "x²" is positive. So the parabola is open upward.

Step 3 :

b² - 4ac = (5)² - 4(1)(6) = 25 - 24 = 1

b² - 4ac = 1 ≥ 0

So, the parabola intersects x - axis.

Step 4 :

To find "x" co-ordinate of the vertex,

let us use the formula x = -b/2a.

x = -b/2a = -5/2(1) = -5/2 = -2.5 --------> x = -2.5

When we plug x = -2.5 in y = x² + 5x + 6, we get y = -0.25

Then the vertex (x,y) = (-2.5, -0.25)

Step 5 :

We can use the results of step 2, step 3, step 4 and sketch the parabola.

Step 6 :

The given inequality is,  

x² + 5x + 6 ≥ 0 --------> y 0 -----> ("y" is positive)

When we look at the above graph, "y" is positive in the intervals 

(-∞, -3] &  [-2,+∞) of "x". 

Hence the solution is, 

x ∈  (-∞, -3] U [-2,+∞)

Let us look at the next problem on "Solving quadratic inequalities graphically".

Example 2 :

Solve the quadratic inequality given below graphically.

-x² + 2x - 5 ≤ 0

Solution :

Step 1 :

Let y = -x² + 2x - 5

Then we have, -x² + 2x - 5 0 --------> y 0

Step 2 :

The sign of  "x²" is negative. So the parabola is open downward.

Step 3 :

b² - 4ac = (2)² - 4(-1)(-5) = 4 - 20 = -16

b² - 4ac = -16 < 0

So, the parabola does not intersects x - axis.

Step 4 :

To find "x" co-ordinate of the vertex,

let us use the formula x = -b/2a.

x = -b/2a = -2/2(-1) = 2/2 = 1 --------> x = 1

When we plug x = 1 in y = -x² + 2x -5, we get y = -4

Then the vertex (x,y) = (1, -4)

Step 5 :

We can use the results of step 2, step 3, step 4 and sketch the parabola.

Step 6 :

The given inequality is,  

-x² + 2x - 5 0 --------> y 0 -----> ("y" is negative)

When we look at the above graph, "y" is negative for real values of "x". 

Hence the solution is, 

All real values  (or) x ∈ R

Let us look at the next problem on "Solving quadratic inequalities graphically".

Example 3 :

Solve the quadratic inequality given below graphically.

-x² + 4 ≥ 0

Solution :

Step 1 :

Let y = -x² + 4

Then we have, -x² + 4 ≥ 0 --------> y 0

Step 2 :

The sign of  "x²" is negative. So the parabola is open downward.

Step 3 :

b² - 4ac = 0 - 4(-1)(4) = 16

b² - 4ac = 16 ≥ 0

So, the parabola intersects x - axis.

Step 4 :

To find "x" co-ordinate of the vertex,

let us use the formula x = -b/2a.

x = -b/2a = 0/2(-1) = 0 --------> x = 0

When we plug x = 0 in y = -x² + 4, we get y = 4

Then the vertex (x,y) = (0, 4)

Step 5 :

We can use the results of step 2, step 3, step 4 and sketch the parabola.

Step 6 :

The given inequality is,  

-x² + 4 ≥ 0 --------> y 0 -----> ("y" is positive)

When we look at the above graph, "y" is positive in the interval 

[-2, 2] of "x". 

Hence the solution is, 

x ∈  [-2, 2] (or) -2 ≤ x ≤ 2

Let us look at the next problem on "Solving quadratic inequalities graphically".

Example 4 :

Solve the quadratic inequality given below graphically.

x² + 4x +7  0

Solution :

Step 1 :

Let y = x² + 4x +7

Then we have, x² + 4x +7  0 --------> y 0

Step 2 :

The sign of  "x²" is positive. So the parabola is open upward.

Step 3 :

b² - 4ac = (4)² - 4(1)(7) = 16 - 28 = -12

b² - 4ac = -12 < 0

So, the parabola does not intersects x - axis.

Step 4 :

To find "x" co-ordinate of the vertex,

let us use the formula x = -b/2a.

x = -b/2a = -4/2(1) = -4/2 = -2 --------> x = -2

When we plug x = -2 in y = x² + 4x +7, we get y = 3

Then the vertex (x,y) = (-2, 3)

Step 5 :

We can use the results of step 2, step 3, step 4 and sketch the parabola.

Step 6 :

The given inequality is,  

x² + 4x +7  0 --------> y 0 ("y" is negative)

When we look at the above graph, "y" is positive for all real values of "x".  

It contradicts the given inequality.

Hence, there is no solution.

You can also visit the following web pages.

Solving quadratic inequalities algebraically

Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equation

Relationship between zeros and coefficients of a quadratic polynomial

Solving absolute value equations

Solving absolute value equations worksheet pdf

Solving absolute value inequalities

Graphing absolute value functions

Greatest integer function

Graphing greatest integer function

Apart from the above stuff example problems on "Solving quadratic inequalities graphically", if you want to know more about "Solving quadratic inequalities graphically", please clcik here.

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