# SOLVING QUADRATIC  INEQUALITIES GRAPHICALLY

Solving Quadratic Inequalities Graphically :

In this section, we will learn, how to solve quadratic inequalities graphically.

## Solving Quadratic Inequalities Graphically - Steps

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.

## Solving Quadratic Inequalities Graphically - Examples

Example 1 :

Solve the quadratic inequality given below graphically.

x2 + 5x + 6 ≥ 0

Solution :

Step 1 :

Let

y  =  x2 + 5x + 6 -----(1)

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)

b2 - 4ac  =  25 - 24

b2 - 4ac  =  1 ≥ 0

So, the parabola intersects x - axis.

Step 4 :

To find x co-ordinate of the vertex, use the formula

x  =  -b / 2a

Substitute 5 for b and 1 for a.

x  =  -b / 2a

x  =  -5 / 2(1)

x  =  -5 / 2

x  =  -2.5

Substitute -2.5 for x in (1)

(1)-----> y  =  (-2.5)2 + 5(-2.5) + 6

y  =  6.25 - 12.5 + 6

y  =  -0.245

Therefore, the vertex is (-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,

(-∞, -3] U [-2,+∞)

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)

b2 - 4ac  =  4 - 20

b2 - 4ac  =  -16 < 0

So, the parabola does not intersects x - axis.

Step 4 :

To find x co-ordinate of the vertex, use the formula

x  =  -b / 2a

Substitute 2 for b and -1 for a.

x  =  -2/2(-1)

x  =  -2/(-2)

x  =  1

Substitute 1 for x in (1).

(1)-----> y  =  -(1)2 + 2(1) - 5

y  =  -1 + 2 - 5

y  =  -4

Therefore, the vertex is (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 all real values of x.

Hence, the solution is,

All real values  (or) x ∈ R

Example 3 :

Solve the quadratic inequality given below graphically.

-x2 + 4 ≥ 0

Solution :

Step 1 :

Let

y  =  -x2 + 4 -----(1)

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)2 - 4(-1)(4)

b2 - 4ac  =  0 + 16

b2 - 4ac  =  16 ≥ 0

So, the parabola intersects x - axis.

Step 4 :

To find x co-ordinate of the vertex, use the formula

x  =  -b / 2a

Substitute 0 for b and -1 for a.

x  =  0 / 2(-1)

x  =  0

Substitute 0 for x in (1)

(1)-----> y  =  -(0)2 + 4

y  =  0 + 4

y  =  4

Therefore, the vertex is (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,

[-2, 2]

Example 4 :

Solve the quadratic inequality given below graphically.

x2 + 4x +7 ≤ 0

Solution :

Step 1 :

Let

y  =  x2 + 4x +7 -----(1)

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)

b2 - 4ac  =  16 - 28

b2 - 4ac  =  -12

So, the parabola does not intersects x - axis.

Step 4 :

To find x co-ordinate of the vertex, use the formula

x  =  -b / 2a

Substitute 4 for b and 1 for a.

x  =  -4 / 2(1)

x  =  -4 / 2

x  =  -2

Substitute -2 for x in (1).

(1)-----> y  =  (-2)2 + 4(-2) + 7

y  =  4 - 8 + 7

y  =  3

Therefore, the vertex is (-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. After having gone through the stuff above, we hope that the students would have understood, how to solve quadratic inequalities graphically.

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