Creating Variable Formulas

The variables are mostly present in the algebras and in other functional parts. The formulas in math are fully consists of the variables only. The variable formulas can be created for the various algebraic expressions and to solve the problems. Creating of the variable formulas can be done and it helps us in solving. Now we are going to see about the creating variable formulas.

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Creating variable formulas:

Now we are going to see creating various variable formulas such as follows,

1. (x + y)2 = x2 + 2xy + y2; x2 + y2 = (x+y)2 −2xy

2. (x − y)2 = x2 − 2xy+ y2; x2 + y2 = (x−y)2 + 2xy

3. (x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + xz)

4. (x + y)3 = x3 + y3 + 3xy(x + y); x3 + y3 = (x + y)3 −3xy(x + y)

5. (x − y)3 = x3 − y3 − 3xy(x − y); x3 − y3 = (x - y)3 + 3xy(x − y)

6. x2 − y2 = (x+y)(x − y)

7. x3 − y3 = (x−y)(x2 + xy + y2)

8. x3 + y3 = (x+y)(x2 − xy + y2)

9. xn − yn = (x−y)(xn−1 + xn−2y + xn−3y2 + ....... +yn−1)

10. xn = x.x.x....n times

11. xm. xn = xm+n

12. (xm)n = xmn = (xn)m

13. (xy)n = xn.yn

14. logx mn = logxm+logx n(if x, m and n are real positive numbers)

15. logx`(m/n)` = logxm − logx n

16. logxmn = nlogxm

17. logxm = logxn, then m=n

Problems in creating variable formulas:

Problem 1:

Solve (3x + 4y)2

Solution:

Now we use the creating formula such as,

(x + y)2 = x2 + 2xy + y2

Let substitute the given expression in the formula,

(3x + 4y)2 = (3x)2 + 2(3x)(4y) + (4y)2

= 9x2 + 24xy + 16y2


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Problem 2:

Solve the expression: (2x + 5y)3

Solution:

Now we are going to solve the expression using the formula

(x + y)3 = x3 + y3 + 3xy(x + y)

Now we will substitute the values as follows,

(2x + 5y)3 = (2x)3 + (5y)3 + 3(2x)(5y)(2x + 5y)

= 8x3 + 125y3 + 30xy (2x + 5y)