HOW TO DO ALGEBRA - SPECIAL PRODUCTS, EXAMPLES, EXERCISE, USING FORMULAS

Your Ad Here





Please study
Algebra Formulas before How To Do Algebra
if you have not already done so.
There we have listed out all the
Algebra Formulas that need to be
remembered.

Also Please study
Proofs of the first six Formulas

Proofs of the Last six Formulas

if you have not already done so.

Here we present Solved Examples
and Exercise problems
on application of those Formulas.

Example 1 of How To Do Algebra

Find the quotient (27x3 - 64)÷(9x2 + 12x + 16) without actual division.

Solution to Example 1 of How To Do Algebra :
Let P = (27x3 - 64)÷(9x2 + 12x + 16)
We know 27 = 3 x 3 x 3 = 33 ⇒27x3 = 33x3 = (3x)3
64 = 4 x 4 x 4 = 43
∴ Numerator of P = (27x3 - 64) = (3x)3 - 43
This looks like (a3 - b3) with (3x) in place of a and (4) in place of b
Denominator of P = (9x2 + 12x + 16) = (3x)2 + (3x)(4) + 42
This looks like (a2 + ab + b2) with (3x) in place of a and (4) in place of b
We have (a3 - b3) = (a - b)(a2 + ab + b2) (See Formula 5)
⇒ (a3 - b3)÷(a2 + ab + b2) = (a - b)
P is like the L.H.S. of this with (3x) in place of a and (4) in place of b
∴ P = (3x - 4) Ans.

Example 2 of How To Do Algebra

If 2x + 3y - 4z = 10 and 3xy - 6yz - 4zx = 15, find 4x2 + 9y2 + 16z2.

Solution to Example 2 of How To Do Algebra :
Let P = 4x2 + 9y2 + 16z2 = (2x)2 + (3y)2 + (4z)2
This looks like a2 + b2 + c2
with (2x) in place of a, (3y) in place of b and (4z) in place of c
Then the given data looks like a + b - c = 10 and (1⁄2 )(ab - bc - ca) = 15
We have (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (See Formula 8)
Putting (-c) in place of c, we get
(a + b - c)2 = a2 + b2 + (-c)2 + 2ab + 2b(-c) + 2(-c)a (See Formula 8)
= a2 + b2 + c2 + 2ab - 2bc - 2ca
a2 + b2 + c2= (a + b - c)2 - 2(ab - bc - ca)
Applying this to P, we get
P = (2x)2 + (3y)2 + (4z)2
= (2x + 3y - 4z)2 - 2{(2x)(3y) - (3y)(4z) - (4z)(2x)}
= (2x + 3y - 4z)2 - 2 x 2{(x)(3y) - (3y)(2z) - (4z)(x)}
= (2x + 3y - 4z)2 - 4{(3xy) - (6yz) - (4zx)}
By data, 2x + 3y - 4z = 10 and 3xy - 6yz - 4zx = 15
Using these in P, we get
P = (10)2 - 4(15) = 100 - 60 = 40. Ans.

Great Deals on School & Homeschool Curriculum Books

Example 3 of How To Do Algebra

Solved Example 3 on How To Do Algebra :

Find the quotient
(x3 + 27y3 + 8z3 - 18xyz)÷(x + 3y + 2z)
without actual division.

Solution to Example 3 of How To Do Algebra :
We have a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) [See Formula 9]
⇒ (a3 + b3 + c3 - 3abc)÷(a + b + c) = (a2 + b2 + c2 - ab - bc - ca).....(i)
Let P = (x3 + 27y3 + 8z3 - 18xyz)÷(x + 3y + 2z)
= {x3 + (3y)3 + (2z)3 - 3(x)(3y)(2z)}÷(x + 3y + 2z)
This looks like Equation (i)
with x in place of a, 3y in place of b and 2z in place of c
Applying Equation (i) here, we get
P = {x3 + (3y)3 + (2z)3 - 3(x)(3y)(2z)}÷(x + 3y + 2z)
= {(x)2 + (3y)2 + (2z)2 - (x)(3y) - (3y)(2z) - (2z)(x)}
= {x2 + 9y2 + 4z2 - 3xy - 6yz - 2zx} Ans.

Get The Best Grades With the Least Amount of Effort

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

The secrets will help you absorb, digest
and remember large chunks of information
quickly and easily so you get the best grades
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
giving up your fun, such as TV, surfing the net,
playing video games or going out with friends!

Know more about the Speed Study System.

Example 4 of How To Do Algebra

Solved Example 4 on How To Do Algebra :

Find the products
(i) (x + 1)(x - 3) (ii) (2x + 5)(5x - 3) (iii) (x + 2)(x + 4)(x + 5)
using the Algebra Formulas.

Solution to Example 4 of How To Do Algebra :
(i) Let P = (x + 1)(x - 3)
We have (x + a)(x + b) = x2 + x(a + b) + ab [See Formula 10]
Comparing P with this Formula, a = 1 and b = -3
∴ P = (x + 1)(x - 3) = x2 + x{1 + (-3)} + {(1)(-3)}
= x2 + x{-2} + {(-3)} = x2 - 2x - 3 Ans.

(ii) Let P = (2x + 5)(5x - 3)
We have(ax + b)(cx + d) = acx2 + x(ad + bc) + bd
Comparing P with this Formula, a = 2, b = 5, c = 5 and d = -3
∴ P = (2x + 5)(5x - 3) = (2)(5)x2 + x{(2)(-3) + (5)(5)} + (5)(-3) = 10x2 +19x - 15. Ans.

Great deals on School & Homeschool Curriculum Books and Software

Example 5 of How To Do Algebra

Solved Example 5 on How To Do Algebra :

If l + m + n = 0, prove that l3 + m3 + n3 = 3lmn

Solution to Example 5 of How To Do Algebra :
l + m + n = 0 ⇒ l + m = -n .........(i)
Cubing both sides, we get
(l + m)3 = (-n)3
l3 + m3 + 3lm(l + m) = -n3 [See Formula 6]
Using the value of (l + m) from (i), we get
l3 + m3 + 3lm(-n) = -n3
l3 + m3 - 3lmn = -n3
l3 + m3 + n3 = 3lmn (Proved.)

Exercise : How To Do Algebra

Solve the following problems on How To Do Algebra

  1. Find the quotient (8x3 - 343)÷(2x - 7) without actual division.
  2. If x + 2y + 3z = 12 and x2 + 4y2 + 9z2 = 44, find 2xy + 6yz + 3zx
  3. Find the quotient
    (l3 + 27m3 + 343n3 - 63lmn)÷(l2 + 9m2 + 49n2 - 3lm - 21mn - 7nl)
    without actual division.
  4. Find the products
    (i) (x - 9)(x - 7) (ii) (5x + 2)(7x + 5) (iii) (x - 3)(x - 8)(x + 6)
    using the Algebra Formulas.
  5. If p + q = r, prove that p3 + q3 + 3pqr = r3
For Answers See at the bottom of the page.

Progressive Learning of Math : How To Do Algebra

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.


Answers to Exercise : How To Do Algebra

Answers to Problems on How To Do Algebra :

  1. 4x2 + 14x + 49
  2. 50
  3. l + 3m + 7n
  4. (i) x2 -16x + 63 (ii) 35x2 + 39x + 10 (iii) x3 - 5x2 - 42x + 144

Research-based personalized
Math Help tutoring program :
How To Do Algebra

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!