Algebra Formula for Class 10

Algebra is a significant branch of Mathematics that has a crucial application in the field of engineering and technologies. It deals with symbols called variables that represent a value of any quantity at any instance and the rules for manipulating those symbols.

In order for the student to excel in the topics like Linear Equations in Two Variables, Quadratic Equations,   etc. one must be thorough with the formula of Algebra Class 10.

Maths Formulas For Class 10 Algebra

Class 10 Algebraic Identities

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 + 2yz + 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 – xy – yz − xz)

15. (x + a)(x + b)(x + 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 = 1/2[(x + y)2 + (x – y)2]

Formula for Quadratic Equations

For ax2 + bx + c = 0,

(α, β) =[−b±(b2−4ac)/√2ac

, when α and β are the roots of the equation.

1. If b2 − 4ac > 0, then the quadratic equation has two distinct real roots.

2. If b2 − 4ac < 0, then the quadratic equation has two imaginary roots.

3. If b2 − 4ac = 0, then the quadratic equation has two equal real roots.

Arithmetic Progression Formula

1. nth term of an arithmetic sequence: an = a + (n – 1)d 

2. Sum of n terms of an arithmetic sequence: Sn = n/2 2a+(n–1)d

3. nth term of a geometric sequence: an = a.rn-1

4. Sum of n terms of a geometric sequence: Sn = a (1 – rn)/ (1 – r), r≠1

5. Sum of infinite terms of a geometric sequence: S = a/ (1 – r)

Important keywords

Some important concepts that are used in algebraic formulas which a student should be aware of are listed below:

• An algebraic expression consists of different combinations of constants and variables linked through various operation symbols of addition, subtraction, division and multiplication. (+, -, /, x)

• Different terms make up an algebraic expression

• A term is the product obtained by multiplying two or more factors.

• Coefficient refers to the numerical factor obtained in a term.

• Like terms are terms with the same algebraic factors. Whereas unlike terms are terms that have different algebraic factors. 

• An algebraic expression with just one single term is known as a monomial.

• An algebraic expression with just two terms is called a binomial expression.

• An algebraic expression with more than two terms is called a polynomial expression. 

• When two or more like terms are added, the result is another like term that has a numerical coefficient. This resultant numerical coefficient is equivalent to the sum of all the numerical coefficients of all the respective like terms.

• When two or more like terms are subtracted, the result is another like term that has a numerical coefficient. This resultant numerical coefficient is equivalent to the difference of all the numerical coefficients of all the respective like terms. 

• All the like terms are added together while all the unlike terms are not included and left as they are in an algebraic expression. 

Class 10 Algebraic problems with solved solutions

Listed below are some algebraic problems with their solved solutions for providing a better understanding to the student of how different algebraic formulas work in different problems.

1. Express the following statements in the form of algebraic expressions using respective variables, constants and arithmetic operations.

1. Subtraction of x from y.

2. One-third of the sum of numbers a and b.

3. The number r multiplied by itself.

4. One-half of the product of numbers s and t.

5. Numbers c and d both squared and subtracted.

6. Number 10 added to five times the product of number v and w

7. Product of numbers y and x added to 20.

8. Sum of numbers a and b and c subtracted from their respective product.

Answer:

1.  y – x

2.  1/3 (a +b)

3.  r2

4.  ½ (st)

5.  c2 – d2 

6.  10 + 5 (vw)

7.  xy + 20

8.  abc – (a + b + c)

2. Find the roots of the equation x2 – 10x – n (n + 10) = 0, where n is a constant.

Answer: 

Given to us is,

x2 – 10x – n (n + 3) = 0

Upon comparing it with the general formula of algebraic expression ax2 + bx + c = 0, we get:

a = 1, b = -10, and c = – n (n + 10)

we also know that, D = (b2 – 4ac)

So

D = (- 10)2 – 4(1) [-n(n + 10)]

= 100 + 4n (n + 10)

= 4n2 + 40n + 100

= (2n + 10)2

substituting the value of D in the quadratic formula: (α, β) = [–b ± √D]/2ac, we get:

(α, β) = [–10 ± √(2n + 10)2]/2x1x- n (n + 10)

Solving this equation, we get

(α, β) = (n + 10, -n)

So, the roots of the given quadratic equation are (n + 10) and -n.

Practice problems for students

1. Find the roots of the equation y2 – 5y – p (p + 5) = 0, where p is a constant.

2. If 1 is a root of the equations az2 + az + 16 = 0 and x2 + x + b = 0, then find the value of zx.

3. Find the value of p so that the quadratic equation px(x – 4) + 16 = 0 has two similar roots.

4. Find the value of q so that the quadratic equation qx (x – 10) + 100 = 0 has two similar roots.

5. Simplify the algebraic expression -2(x – 5) + 4(-3 x + 8)

6. Expand and simplify the algebraic x + 2) (x – 2) – (-x – 4)

7. For what values of k the equation -x2 + 3kx – 9 = 0 has one real solution?

8. If (p2 – q2) = 10 and (p + q) = 2, find p and q.

Formula for Linear Equation in Two Variables

Let the pair of linear equations in two variables be:

a1x + b1 + c1 = 0 and a2x + b2 + c2= 0

When a1, b1, c1, & a2, b2, c2 are real numbers where a12+b12 ≠ 0 & a22 + b22 ≠ 0

There are infinite possible solutions for the above equation.

Did You Know?

Complex forms of mathematics as Algebra were used by Ancient Egyptians back in 3000BC for equations to find the approximate area of circles.