QUADRATIC FUNCTION - QUADRATIC INEQUALITIES, SOLVED EXAMPLES, EXERCISES

Your Ad Here





Please study

Formulas of Quadratic Inequalities before applying them in Quadratic Function,
if you have not already done so.

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 - 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

Solve the quadratic inequalities
(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.

Great Deals on School & Homeschool Curriculum Books

Solved Example 3 : Quadratic Function

Solved Example 3 on Quadratic Function :

Solve the quadratic inequalities
(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 Great deals on School & Homeschool Curriculum Books and Software

Exercise : Quadratic Function

Solve the following problems on Quadratic Function :

  1. Solve the quadratic inequalities
    (i) 16x - 15 - 4x2 > 0 (ii) 16x - 15 - 4x2 ≥ 0
  2. Solve the quadratic inequalities
    (i) 16x - 15 - 4x2 < 0 (ii) 16x - 15 - 4x2 ≤ 0
  3. Solve the quadratic inequalities
    (i) 6x - 1 - 9x2 > 0 (ii) 6x - 1 - 9x2 ≥ 0

For Answers see at the bottom of the page.

Progressive Learning of Math : Quadratic Function

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.


Research-based personalized
Math Help tutoring program :
Quadratic Function

Here is a resource for Solid Foundation in
Math Fundamentals from Middle thru High School.
You can check your self by the

FREE TRIAL.

Are you spending lot of money for math tutors to your
child and still not satisfied with his/her grades ?

Do you feel that more time from the tutor and
more personalized Math Help to identify and fix
the problems faced by your child will help ?

Here is a fool proof solution I strongly recommend
and that too With a minuscule fraction of the amount
you spent on tutors with unconditional 100% money
back Guarantee, if you are not satisfied.

SUBSCRIBE, TEST, IF NOT SATISFIED, RETURN FOR FULL REFUND

It is like having an unlimited time from an excellent Tutor.

It is an Internet-based math tutoring software program
that identifies exactly where your child needs help and
then creates a personal instruction plan tailored to your
child’s specific needs.

If your child can use a computer and access
the Internet, he or she can use the program.
And your child can access the program anytime
from any computer with Internet access.

Unique program to help improve math skills quickly and painlessly.

There is an exclusive, Parent Information Page provides YOU
with detailed reports of your child’s progress so you can
monitor your child’s success and give them encouragement.
These Reports include

  • Time spent using the program
  • Assessment results
  • Personalized remediation curriculum designed for your child
  • Details the areas of weakness where your child needs additional help
  • Provides the REASONS WHY your child missed a concept
  • List of modules accessed and amount of time spent in each module
  • Quiz results
  • Creates reports that can be printed and used to discuss issues with your child’s teachers
These reports are created and stored in a secure section
of the program, available exclusively to you, the parent.
The section is accessed by a password that YOU create and use.
No unauthorized users can access this information.

Personalized remediation curriculum designed for your child

Thus The features of this excellent Tutoring program are

  • Using detailed testing techniques
  • Identifing exactly where a student needs help.
  • Its unique, smart system pinpointing precise problem areas -
  • slowly and methodically guiding the student
  • raising to the necessary levels to fix the problem.

Not a “one-size-fits-all” approach!

Its research-based results have proven that
it really works for all students! in improving
math skills and a TWO LETTER GRADE INCREASE in
math test scores!,if they invest time in using
the program.

Proven for More than 10,000 U.S. public school
students who increased their math scores.

Proven methodology!



Answers to Exercise : Quadratic Function

Solution to the problems on Quadratic Function :

(1) (i) 3⁄2 < x < 5⁄2. (ii) 3⁄2 ≤x ≤ 5⁄2
(2) (i) 3⁄2 > x > 5⁄2. (ii) 3⁄2 ≥x ≥ 5⁄2
(3) (i) no solution (ii) 1⁄3