There we presented 6 Formulas of Quadratic Inequalities with Proofs. 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

Solve the quadratic inequalities (i) 2 - 5x - 18x^{2} > 0 (ii) 2 - 5x - 18x^{2} ≥ 0

Solution to Solved Example 1 of Quadratic Function :

Solving (i) : STEP 1 : The given inequation is 2 - 5x - 18x^{2} > 0 ⇒ -18x^{2} - 5x + 2 > 0 Dividing both sides by -18, we get x^{2} + (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 x^{2} + (5⁄18)x - 2⁄18 = 0:

Comparing the L.H.S. with ax^{2} + bx + c, we get a = 1, b = (5⁄18) and c = -2⁄18 Discriminant = Δ = b^{2} - 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

∴x^{2} + (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 - 18x^{2} > 0 is x ∈ (-1⁄2, 2⁄9) or -1⁄2 < x < 2⁄9. Ans.

(ii) To solve 2 - 5x - 18x^{2} ≥ 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 - 18x^{2} ≥ 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

Solve the quadratic inequalities (i) 2 - 5x - 18x^{2} < 0 (ii) 2 - 5x - 18x^{2} ≤ 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 - 18x^{2} < 0 ⇒ -18x^{2} - 5x + 2 < 0 Dividing both sides by -18, we get x^{2} + (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 x^{2} + (5⁄18)x - 2⁄18 = 0: From Solved Example 1 above, x = 2⁄9 or -1⁄2

∴x^{2} + (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 - 18x^{2} < 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 - 18x^{2} ≤ 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 - 18x^{2} ≤ 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.

Solve the quadratic inequalities (i) 4x - 1 - 4x^{2} > 0 (ii) 4x - 1 - 4x^{2} ≥ 0

Solution to Solved Example 3 of Quadratic Function :

Solving (i) : STEP 1 : The given inequation is 4x - 1 - 4x^{2} > 0 ⇒ -4x^{2} + 4x - 1 > 0 Dividing both sides by -4, we get x^{2} - x + 1⁄4 < 0 [As we divided with negative number, '>' became '<'. See Property 3 of Algebra Inequalities.]

STEP 2 : To find the roots of x^{2} - x + 1⁄4 = 0 : Comparing the L.H.S. with ax^{2} + bx + c, we get a = 1, b = -1 and c = 1⁄4 Discriminant = Δ = b^{2} - 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

∴ x^{2} - 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. ∴ x^{2} - x + 1⁄4 < 0 has no real solution for x.

Thus, 4x - 1 - 4x^{2} > 0has no solution in the set of Real numbers. Ans.

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

Solve the following problems on Quadratic Function :

Solve the quadratic inequalities (i) 16x - 15 - 4x^{2} > 0 (ii) 16x - 15 - 4x^{2} ≥ 0

Solve the quadratic inequalities (i) 16x - 15 - 4x^{2} < 0 (ii) 16x - 15 - 4x^{2} ≤ 0

Solve the quadratic inequalities (i) 6x - 1 - 9x^{2} > 0 (ii) 6x - 1 - 9x^{2} ≥ 0

For Answers see at the bottom of the page.

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