The following shortcuts will be useful to solve quadratic inequalities.

Shortcut 1 : Other Cases : If both (ax2 + bx + c) and x2 have same signs, then there is solution for the given inequality.

If (ax2 + bx + c) and x2 have different signs,we have to find the value of (b2 - 4ac). Then we can know whether there is solution or not by using the hints given on the above tables.

From shortcut 1, if there is solution for the given quadratic inequality, then follow shortcut 2 to know the type of solution.

Shortcut 2 : Example 1 :

x2 + 5x + 6 ≥ 0

Solution :

Both (x2 + 5x +6) and x2 have same signs (positive). So, there is solution for the given inequality.

To know the type of solution, let us find the value of

b2 -4ac

b2 - 4ac  =  52 - 4(1)(6)

b2 - 4ac  =  25 - 24

b2 - 4ac  =  1

We get,

b2 - 4ac  =  1 -----> b2 - 4ac  ≥  0

Therefore, there is interval solution.

Let us find the interval solution.

Let x2 + 5x + 6  =  0  and solve for x.

x2 + 5x + 6  =  0

(x + 2)(x + 3)  =  0

x  =  -2  or   x  =  -3

Let us mark x  =  -2  and  x  =  -3 on the real number line. Testing the interval (-∞, -3] :

Take a random value in the interval (-∞, -3], say

x  =  -4

Substitute -4 for x in the given inequality.

x2 + 5x + 6  =  (-4)2 + 5(-4) + 6

x2 + 5x + 6  =  16 - 20 + 6

x2 + 5x + 6  =  2 ≥ 0

The interval (-∞, -3] satisfies the given inequality.

Testing the interval [-3, -2] :

Take a random value in the interval [-3, -2], say

x  =  -2.5

Substitute -2.5 for x in the given inequality.

x2 + 5x + 6  =  (-2.5)2 + 5(-2.5) + 6

x2 + 5x + 6  =  6.25 - 12.5 + 6

x2 + 5x + 6  =  -0.25 < 0

The interval [-3, -2] does not satisfy the given inequality.

Testing the interval [-2, +∞] :

Take a random value in the interval [-2, +∞), say

x  =  0

Substitute 0 for x in the given inequality.

x2 + 5x + 6  =  (0)2 + 5(0) + 6

x2 + 5x + 6  =  0 - 0 + 6

x2 + 5x + 6  =  6  0

The interval [-2, +∞) satisfies the given inequality.

Hence, the solution is

(-∞, -3] U [-2, +∞)

(or)

x ≤ -3  or  x ≥ -2

Example 2 :

-x² + 2x - 5 ≤ 0

Solution :

Both (-x2 +2x - 5) and x2 have same signs (negative). So, there is solution for the given inequality.

To know the type of solution, let us find the value of

b2 -4ac

b2 - 4ac  =  (2)2 - 4(-1)(-5)

b2 - 4ac  =  4 - 20

b2 - 4ac  =  -16 < 0

Hence, the solution is

all real values

(or)

x ∈ R

Example 3 :

-x2 + 4 0

Solution :

Here, (-x2 + 4) and x2 have different signs.

Let us find the (b2 -4ac).

b2 -4ac  =  (0)2 - 4(-1)(4)

b2 -4ac  =  0 + 16

b2 -4ac  =  16 ≥ 0

(-x2 + 4)  and x2 have different signs and b2 -4ac  ≥ 0.

According to the shortcuts explained above, we have interval solution for the given quadratic inequality.

Let us find the interval solution.

Let -x2 + 4 = 0  and solve for x.

- x2 + 4 = 0

Multiply each side by -1.

x2 - 4 = 0

x2 - 22 = 0

(x + 2)(x - 2)  =  0

x  =  -2  or  x  =  2

Let us mark x = -2  and  x = 2 on the real number line. Testing the interval (-∞, -2] :

Take a random value in the interval (-∞, -2], say

x  =  -3

Substitute -3 for x in the given inequality.

-x2 + 4  =  -(-3)2 + 4

-x2 + 4  =  -9 + 4

-x2 + 4  =  -5 < 0

The interval (-∞, -2] does not satisfy the given inequality.

Testing the interval [-2, 2] :

Take a random value in the interval [-2, 2], say

x  =  0

Substitute 0 for x in the given inequality.

-x2 + 4  =  -(0)2 + 4

-x2 + 4  =  4 ≥ 0

The interval [-2, 2] satisfies the given inequality.

Testing the interval [2, +∞] :

Take a random value in the interval [2, +∞), say

x  =  3

Substitute 3 for x in the given inequality.

-x2 + 4  =  -(3)2 + 4

-x2 + 4  =  -9 + 4

-x2 + 4  =  -5 < 0

The interval [2, +∞) does not satisfy the given inequality.

Hence the solution is

[-2, 2]

(or)

-2 ≤ x ≤ 2

Example 4 :

x2 + 4x +7 0

Solution :

Here, (x2 + 4x + 7) and x2 have different signs.

Let us find the value of (b2 -4ac).

b2 -4ac  =  (4)2 - 4(1)(7)

b2 -4ac  =  16 - 28

b2 -4ac  =  - 12 < 0

(-x2 + 4) and x2 have different signs and b2 -4ac < 0.

As per the shortcuts explained above, there is no solution for the given inequality. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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