Algebra Formula: Important Formulas in Algebra & Solved Examples

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Anjali Mishra

Content Writer-SME | Updated On - Oct 24, 2024

Alphabetical notation in any equation of mathematics and science is called algebra formula. It consists of symbols that are known as variables that represent quantity without a fixed value. The most common letters used to express algebraic problems and equations are X, Y, A, and B. Just as an engineer's life mainly revolves around wires, gadgets, current and electricity, similarly for a mathematician, algebra is an important tool, their life also revolves around the same.

Real numbers, complex numbers, matrices, vectors, and other concepts are all part of algebra.​ Algebra formulas have a wide range of applications, which is why it is considered one of the important units. In this article, we will cover all the important algebraic formulas which are used in all classes starting from class 8.


Algebraic Formulas

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Inclusion of new topics in Mathematics is usually seen after class 8, depending on the type of board (CBSE or state or ICSE) and the books run by different schools. Before this, students may be aware of different types of numbers like real numbers, prime numbers etc. but after class 8, they are usually familiar with an important chapter “Algebra” which forms the basis of complex chapters like Trigonometry.

Algebra is a branch of mathematics that deals with both numbers and letters. The value of numbers is fixed and the letters or alphabets represent the unknown quantities in the algebraic formulaA combination of numbers, letters, factorials, matrices, etc. is used to form an algebraic equation or algebraic formula. 

Here is the list of all the important algebraic formulas

  1. (a+b)2 = a2 + 2ab + b2
  2. (a-b)2 = a2 – 2ab + b2
  3. (a+b)(a-b) = a2 – b2
  4. (x +a)(x+b) = x2 +(a+b)x +ab
  5. (x+a)(x-b) = x2 + (a-b)x – ab
  6. (x-a)(x+b) = x2 + (b-a)x – ab
  7. (x-a)(x-b) = x2 – (a+b)x +ab
  8. (a+b)3 = a3 + b3 + 3ab(a+b)
  9. (a-b)3 = a3 – b3 – 3ab (a-b)
  10. (x+y+z)2 = x2 + y2 +z2 + 2xy + 2yx + 2xz
  11. (x+y-z)2 = x2 + y2 +z2 + 2xy – 2yz – 2xz
  12. (x – y +z)2 = x2 + y2 +z2  - 2xy – 2yz +2xz 
  13. (x-y-z)2 = x2 + y2 +z2  - 2xy +2yz -2xz
  14. X3 + y3 +z3 – 3xyz = (x+y+z) (x2+ y2 +z2)
  15. (x+a)(a+b)(a+c) = x3 + (a+b+c) x2 + (ab +bc+ ca)x + abc
  16. X3 + y3 = (x+y) (x2 – xy + y2)
  17. X3 – y3 = (x- y) (x2 + xy + y2)
  18. X2 + y2 + z2 –xy – yz – zx = 12[(x-y)2 + (y-z)2 + (z-x)2
  19. X2 + y2 = ½[(x +y)2 + (x – y)2 ]
  20. a1/x = xa
  21. (a/b)x = a2/b2
  22. (ab)x = axbx
  23. (am)n = amn
  24. am/an = am-n
  25. Laws of Exponents am x an = am+n
  26. a8- b8 = (a4 + b4) (a2 +b2) (a+b)(a-b)
  27. a4 + a2 + 1 = (a2 + a +1)(a2-a + 1)
  28. a4 + a2b2 + b4 = (a2 + ab + b2 )(a2 –ab + b2)
Algebra Formula
Important Algebraic Formulas

Linear Equation in Two Variables

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Algebra formulas are classified into two parts: basic and advanced algebra. Basic algebraic formulas are generally used by students up to class 10. Next, advanced algebraic equations are used to solve complex problems and write equations. Application of advanced level algebraic formulas is also seen in the engineering field.

An equation is said to be a linear equation in two variables if it is written in the form of ax + by + c = 0, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero.

For example: 

  • a1x + b1y + c1 = 0
  • a2x + b2y + c2 = 0

Read More: Arithmetic Progressions Revision Notes


Formulas for Quadratic Equations

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For any quadratic equation, algebraic formula is given as: 

Let an algebraic equation, ax2 + bx + c = 0,

So, the formula for this quadratic equation will be: 

(α,ß) = [-b ± √(b2 – 4ac)]/2ac

  • The quadratic equation will have two distinct real roots if b2- 4ac >0
  • The quadratic equation will have two imaginary roots if b2 – 4ac < 0
  • The quadratic equation has two real equivalent real roots if b2 – 4ac = 0

Progression Formulas

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Progression formulas are given below

  • The nth term of an arithmetic sequence (an) = a + (n-1)d
  • The sum of n terms in an arithmetic sequence (sn) = n/2 [2a + (n-1)d]
  • The nth term of a geometric sequence (an) = a.rn-1
  • The sum of n terms of a geometric sequence (sn) = a(1-rn)/(1-r)
  • The sum of infinite terms of a geometric sequence (s) = a/(1-r)

Things to Remember

  • A single equation without the equal sign is known as an algebraic expression whereas a set of equations with the equal sign is known as the algebraic equation.
  • An equation that has a mixture of variables and is only true for some value of x is known as an algebraic equation.
  • A polynomial of 1 degree is known as linear, 2 degrees is known as quadratic, and degree 3 is known as a cubic polynomial.
  • A quadratic equation has 2-3 zeros which form the x coordinators of the points where the graph of y=p(x) intersects the x-axis.

Sample Questions

Ques. What is algebra? (1 Mark)

Ans. Algebra is the study of variables and the rules that govern their use in formulae.

Ques. William had a few chocolates with him. Jack came across and took away five of his chocolates. And then he had just seven chocolates remaining with him. How many chocolates did he have before Jack came to him? (2 Marks)

Ans. Let William have x chocolates with him, and Jack take away five of his chocolates, then

x − 5 = 7

Hence, we subtract five from x. Moreover, after all this, William was left with seven chocolates. This entire puzzle will then equal seven. 

x − 5 + 5 = 7 + 5

x = 12

Ques. 43 x 42 = ? (2 Marks)

Ans. Using the exponential formula (am)(an) = am+n
where a = 4
⇒ 43 x 42  
= 43+2
= 45
= 1024

Ques. Calculate: 172 – 42 (2 Marks)

Ans. We know that: 

a2 – b2 = (a+b)(a-b)

So, 

172 – 42 = (17+4)(17-4) 

= 13 x 21

=273

Ques. Simplify the expression: 12m2 -9m +5m -4m2 -7m +10 (2 Marks)

Ans. Given that: 

12m2 – 4m2 +5m -9m -7m +10

= (12-4)m2 + (5-9-7)m+10

=8m2 + (-4-7)m + 10

= 8m2 + 11m + 10

Ques. If the quadratic equation is mx(x-7) + 49 = 0. Find the value of m when the equation has two equal roots. (3 Marks)

Ans. mx (x-7)+49 = 0

mx2 – 7mx + 49 = 0

a= m, b= -7m, c = 49

D = b2 -4ac = 0

= (-7m)2 – 4 (m) (49) = 0

= 49m2 – 4m (49) = 0

= 49m (m-4) = 0

= 49m = 0 or m – 4 = 0

m = 0 (rejected) or m = 4

m= 4

Ques. If the quadratic equation is x2 – 3x – m (m +3 ) = 0 and the value of m is constant. Find the root of the equation. (3 Marks)

Ans. x2 – 3x – m(m+3) = 0

a = 1, b= -3, c = -m(m+3) 

D = (b2 – 4ac)

So,

D= (-3)2 -4 (1) = 9 +4m(m+3)

= 4m2 +12m +9

= (2m+3)2

(α,ß) = [-3±√(2m+3)2]/2x1x-m(m+3)

= (α,ß) = (m+3,-m)

Ques. The equation ay2+ay+3 = 0 and y2+y+b= 0 has the root 1. Calculate the value of ab. (3 Marks)

Ans. Given that: 

ay2+ay+3 = 0 … (i)

y2+y+b= 0 … (ii)

By putting the value of y = 1,

a(1)2 + a(1) +3 = 0

2a = -3

a = -3/2

by putting the value of y =1 in the equation y2 +y+b =0,

12 + 1+b = 0

b = -2

therefore, ab =(-3/2)(-2) = 3

Ques. Find the value of p if the quadratic equation px2 -(25 - √p)x + 15 = 0 which has two equal roots. (3 Marks)

Ans. px2 -25 - √px + 15 = 0

a = p , b = -25- √p , c = 15

D = 0 (for equal roots)

D = b2 – 4ac -0

0 = 4 x 5p2 – 60p

0 = 20p2 – 60p = 20p2 = 60p

p = 60p 20p = 3

Therefore, p = 3

Ques. Solve the equation: 5x +3y = 9 and 2x - 3y = 12. (3 Marks)

Ans. Given that: 

5x +3y = 9 … (i)

2x - 3y = 12 … (ii)

By adding equation (i) and (ii), we get

7x = 21

x = 3

By substituting the value of x in equation (i), we get

5 (3) + 3y = 9

15 + 3y = 9

3y = 9-15 = -6

Y = -2

Thus, (x,y) = (3,-2)

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CBSE X Related Questions

1.

Form the pair of linear equations for the following problems and find their solution by substitution method.

(i) The difference between two numbers is 26 and one number is three times the other. Find them.

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km.

(v) A fraction becomes\(\frac{ 9}{11}\), if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes \(\frac{5}{6}\). Find the fraction.

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

      2.

      Solve the following pair of linear equations by the substitution method. 
      (i) x + y = 14 
          x – y = 4   

      (ii) s – t = 3 
          \(\frac{s}{3} + \frac{t}{2}\) =6 

      (iii) 3x – y = 3 
            9x – 3y = 9

      (iv) 0.2x + 0.3y = 1.3 
           0.4x + 0.5y = 2.3 

      (v)\(\sqrt2x\) + \(\sqrt3y\)=0
          \(\sqrt3x\) - \(\sqrt8y\) = 0

      (vi) \(\frac{3x}{2} - \frac{5y}{3}\) =-2,
          \(\frac{ x}{3} + \frac{y}{2}\) = \(\frac{ 13}{6}\)

          3.
          If 3 cot A = 4, check whether \(\frac{(1-\text{tan}^2 A)}{(1+\text{tan}^2 A)}\) = cos2 A – sinA or not

              4.
              A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

                  5.
                  A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

                      6.
                      Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) \(x + y = 5\),\( 2x + 2y = 10\) (ii)\( x – y = 8 , 3x – 3y = 16\) (iii) \(2x + y – 6 = 0\) , \(4x – 2y – 4 = 0\) (iv) \(2x – 2y – 2 = 0,\) \( 4x – 4y – 5 = 0\)

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