if you have not already done so.

There we presented 6 Formulas of
The method of Solving Quadratic Inequalities
is also presented Step wise.

Here we deal with Applying those Formulas
and the method in Solving Problems.

## Solved Example 1 : Quadratic Function

(i) 2 - 5x - 18x2 > 0 (ii) 2 - 5x - 18x2 ≥ 0

Solution to Solved Example 1 of Quadratic Function :

Solving (i) :
STEP 1 :
The given inequation is 2 - 5x - 18x2 > 0
⇒ -18x2 - 5x + 2 > 0
Dividing both sides by -18, we get
x2 + (5⁄18)x - 2⁄18 < 0
[As we divided with negative number, '>' became '<'.
See Property 3 of Linear Inequalities.]

STEP 2 :
To find the roots of x2 + (5⁄18)x - 2⁄18 = 0:

Comparing the L.H.S. with ax2 + bx + c, we get
a = 1, b = (5⁄18) and c = -2⁄18
Discriminant = Δ = b2 - 4ac
= (5⁄18)2 - 4(1)(-2⁄18) = {25 + 8(18)}⁄{(18)(18)}
= (169)⁄(324) > 0 ⇒ the roots are real and distinct.

√Δ = √{(169)⁄(324)} = 13⁄18
By quadratic formula, the roots of the equation are given by
x = (-b ± √Δ)⁄(2a)
= {-(5⁄18) ± (13⁄18)}⁄{2(1)} = {(-5+13)⁄18}⁄2 or {(-5-13)⁄18}⁄2
= (8⁄18)⁄2 or (-18⁄18)⁄2 = 2⁄9 or -1⁄2

x2 + (5⁄18)x - 2⁄18 = {x - (2⁄9)}{x - (-1⁄2)}

STEP 3 :
By Formula 2 above,{x - (2⁄9)}{x - (-1⁄2)} < 0
x lies between (-1⁄2) and (2⁄9) or x ∈ (-1⁄2, 2⁄9) or -1⁄2 < x < 2⁄9

Thus, the solution of the given inequation 2 - 5x - 18x2 > 0 is
x ∈ (-1⁄2, 2⁄9) or -1⁄2 < x < 2⁄9. Ans.

(ii) To solve 2 - 5x - 18x2 ≥ 0.

The solution of (ii) is slightly different from that of (i).
Put '≤' in place of '<' in the above solution.
Thus, the solution of 2 - 5x - 18x2 ≥ 0 becomes
-1⁄2 ≤ x ≤ 2⁄9 or x ∈ [-1⁄2, 2⁄9]
When we put square brackets in place of circular brackets,
the extreme values are included in the range.

## Solved Example 2 : Quadratic Function

Solved Example 2 on Quadratic Function

(i) 2 - 5x - 18x2 < 0 (ii) 2 - 5x - 18x2 ≤ 0

Solution to Solved Example 2 of Quadratic Function :

Solving (i) :
The same quadratic polynomial as in solved example (i)
is taken withinequality sign reversed.
STEP 1 :
The given inequation is 2 - 5x - 18x2 < 0
⇒ -18x2 - 5x + 2 < 0
Dividing both sides by -18, we get
x2 + (5⁄18)x - 2⁄18 > 0
[As we divided with negative number, '<' became '>'.
See Property 3 of Algebra Inequalities.]

STEP 2 :
To find the roots of x2 + (5⁄18)x - 2⁄18 = 0:
From Solved Example 1 above, x = 2⁄9 or -1⁄2

x2 + (5⁄18)x - 2⁄18 = {x - (2⁄9)}{x - (-1⁄2)}

STEP 3 :
By Formula 1 above, {x - (2⁄9)}{x - (-1⁄2)} > 0
x does not lie between (-1⁄2) and (2⁄9) or
x ∉ (-1⁄2, 2⁄9) or x ∈ (-∞,-1⁄2) ∪ (2⁄9, ∞) or -1⁄2 > x > 2⁄9

Thus, the solution of the given inequation 2 - 5x - 18x2 < 0 is
x ∉ (-1⁄2, 2⁄9) or x ∈ (-∞,-1⁄2) ∪ (2⁄9, ∞) or -1⁄2 > x > 2⁄9. Ans.

(ii) To solve 2 - 5x - 18x2 ≤ 0

The solution of (ii) is slightly different from that of (i).
Put '≥' in place of '>' in the above solution.
The solution of 2 - 5x - 18x2 ≤ 0 becomes
-1⁄2 ≥ x ≥ 2⁄9 or x ∈ (-∞,-1⁄2] ∪ [2⁄9, ∞)
When we put square brackets in place of circular brackets,
the extreme values are included in the range.

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## Solved Example 3 : Quadratic Function

Solved Example 3 on Quadratic Function :

(i) 4x - 1 - 4x2 > 0 (ii) 4x - 1 - 4x2 ≥ 0

Solution to Solved Example 3 of Quadratic Function :

Solving (i) :
STEP 1 :
The given inequation is 4x - 1 - 4x2 > 0
⇒ -4x2 + 4x - 1 > 0
Dividing both sides by -4, we get
x2 - x + 1⁄4 < 0
[As we divided with negative number, '>' became '<'.
See Property 3 of Algebra Inequalities.]

STEP 2 :
To find the roots of x2 - x + 1⁄4 = 0 :
Comparing the L.H.S. with ax2 + bx + c, we get
a = 1, b = -1 and c = 1⁄4
Discriminant = Δ = b2 - 4ac = (-1)2 - 4(1)(1⁄4) = {1 - 1} = 0
⇒ the roots are real and equal.
By quadratic formula, the roots of the equation are given by
x = (-b ± √Δ)⁄(2a) = {-(-1) ± 0}⁄{2(1)} = 1⁄2 or 1⁄2

x2 - x + 1⁄4 = {x - (1⁄2)}2

STEP 3 :
By Formula 4 above,{x - (1⁄2)}2 < 0
x ∉ R (the real number set) or x ∉ (-∞, ∞)
i.e. x can not any take any real value.
x2 - x + 1⁄4 < 0 has no real solution for x.

Thus, 4x - 1 - 4x2 > 0has no solution in the set of Real numbers. Ans.

(ii) To solve 4x - 1 - 4x2 ≥ 0
We have seen 4x - 1 - 4x2 = 0 has the solution x = 1⁄2
and 4x - 1 - 4x2 > 0 has no solution.
∴ the solution of 4x - 1 - 4x2 ≥ 0 is {1⁄2}.Ans.

NOTE: the brackets { } are used for set of individual elements.
∴ {1⁄2} represents single element 1⁄2.

The First three Formulas are applied
in the above Three Problems.

For Problems which deal with
the next three Formulas, go to

Set 2 of Problems on Quadratic Inequalities

Solve the following problems on Quadratic Function :

(i) 16x - 15 - 4x2 > 0 (ii) 16x - 15 - 4x2 ≥ 0
(i) 16x - 15 - 4x2 < 0 (ii) 16x - 15 - 4x2 ≤ 0
(i) 6x - 1 - 9x2 > 0 (ii) 6x - 1 - 9x2 ≥ 0

For Answers see at the bottom of the page.

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