# SOLVE ALGEBRA PROBLEMS USING FORMULAS, EXAMPLES, EXERCISE

Algebra Formulas before Solve Algebra Problems
if you have not already done so.
There we have listed out all the
Algebra Formulas that need to be
remembered.

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 Solve Algebra Problems

Solve Algebra Problems given below.

Find (i) (100.5)2 (ii) (100.5)3 using Algebra Formulas

Solution to Example 1 of Solve Algebra Problems :
(i) Let P = (100.5)2 = (100 + 0.5)2
This looks like (a + b)2 with (100) in place of a and (0.5) in place of b
We have (a + b)2 = a2 + 2ab + b2 (See Formula 1)
∴ P = (100 + 0.5)2 = (100)2 + 2(100)(0.5) + (0.5)2
= 10000 + 100 + 0.25 = 100(100 + 1) +0.25
= 10100 + 0.25 = 10100.25.Ans.

(ii) Let P = (100.5)3 = (100 + 0.5)3
This looks like (a + b)3
with (100) in place of a and (0.5) in place of b
We have (a + b)3 = a3 + 3a2b + 3ab2 + b3 (See Formula 7)
∴ P = (100 + 0.5)3 = 1003 + 3(100)2(0.5) + 3(100)(0.5)2 + (0.5)3
= (100)2 (100 + 1.5) + 75 + 0.125 = (101.5)10000 + 75.125
= 1015000 + 75.125 = 1015075.125 Ans.

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## Example 2 of Solve Algebra Problems

Solve Algebra Problems given below.

Find (2x - 3y)(2x + 3y)(4x2 + 9y2) and verify the result for x = -1, y = 1.

Solution to Example 2 of Solve Algebra Problems :
Let P = (2x - 3y)(2x + 3y)(4x2 + 9y2)
(2x - 3y)(2x + 3y) looks like (a - b)(a + b) with 2x in place of a and 3y in place of b
We have (a - b)(a + b) = (a + b)(a - b) = a2 - b2 (See Formula 3)
∴ (2x - 3y)(2x + 3y) = (2x)2 - (3y)2 = (4x2 - 9y2)
Using this in P, we get
P = (4x2 - 9y2)(4x2 + 9y2)
This again looks like (a - b)(a + b)
with 4x2 in place of a and 9y2 in place of b
So, Applying Formula 3 again, we get
P = (4x2)2 - (9y2)2 = 16x4 - 81y4. Ans.

Verifying the result for x = -1, y = 1 :

We have (2x - 3y)(2x + 3y)(4x2 + 9y2) = (16x4 - 81y4)
For x = -1, y = 1, L.H.S. = (2x - 3y)(2x + 3y)(4x2 + 9y2)
= {2(-1) - 3(1)}{2(-1) + 3(1)}{4(-1)2 + 9(1)2)
= {-5}{1}{4 + 9} = -5 x 13 = -65
For x = -1, y = 1, R.H.S. = {16(-1)4 - 81(1)4)}= 16 - 81 = -65
∴ L.H.S. = R.H.S. (Verified.)

## Example 3 of Solve Algebra Problems

Solve Algebra Problems given below.

Find 109 x 91 using Algebra Formula

Solution to Example 3 of Solve Algebra Problems :
As you can see, the given product is
product of sum and difference of two numbers,
to which we can apply Formula 3.
Let P = 109 x 91 = (100 + 9)(100 - 9)
We have (a + b)(a - b) = a2 - b2 (See Formula 3)
∴ P = (100 + 9)(100 - 9) = (100)2 - (9)2 = 10000 - 81 = 9919. Ans.

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## Example 4 of Solve Algebra Problems

Solve Algebra Problems given below.

Find by what expression we should multiply (16x2 - 20xy + 25y2)
to get the result as sum of two cubes.

Solution to Example 4 of Solve Algebra Problems :
Let P = (16x2 - 20xy + 25y2) = (4x)2 - (4x)(5y) + (5y)2
This looks like (a2 - ab + b2) with (4x) in place of a and (5y) in place of b
We have(a + b)(a2 - ab + b2) = a3 + b3 (See Formula 4)
⇒ (a2 - ab + b2) is to be multiplied with (a + b)
to get (a3 + b3) which is the sum of two cubes.
∴ P is to be multiplied with (4x + 5y)
to get the result as sum of two cubes. Ans.

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## Exercise : Solve Algebra Problems

Solve Algebra Problems given below :

1. Find (i) (99.5)2 (ii) (99.5)3 using Algebra Formulas
2. Find (p - 4q)(p + 4q)(p2 + 16q2)
and verify the result for p = 2, q = 1.
3. Find (30.5) x (29.5) using Algebra Formula
4. Find the product (2x + 3y)(4x2 - 6xy + 9y2)
without actual multiplication
For Answers See at the bottom of the page.

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## Answers to Exercise : Solve Algebra Problems

1. (i) 9900.25 (ii) 985074.875
2. p4 - 256q4 and -240 = -240
3. 899.75
4. 8x3 + 27y3