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**Algebra formulas for class 10** are provided here for students. The study of Algebra requires an intrinsic and deep understanding of formulas, terms and concept. Hence, to ease this task, we provide to you “Algebra Formulas for class 10”, a thorough and complete guide exclusively drafted to boost the confidence of the students. All algebra formulas and algebraic identities for class 10 in a single page format will help students to grasp and understand every concept thoroughly and solve problems easily.

The CBSE class 10 Mathematics consists of algebra in many chapters like:

- Algebraic method of solving a pair of linear equation- Elimination Method
- Algebraic method of solving a pair of linear equation- Substitution Method
- Algebraic method of solving a pair of the linear equation- Cross Multiplication Method

## List of Algebra Formulas for Class 10

All the algebraic identities for class 10 formulas are provided here. A student may feel difficult to note them and read. To make it easy for them, all the important algebra formulas for class 10 are listed below.

### Algebraic Identities For Class 10

### Linear Equation in Two Variables

- (a + b)
^{2}= a^{2 }+ 2ab + b^{2} - (a − b)
^{2}= a^{2}− 2ab + b^{2} - (a + b)(a – b) = a
^{2}– b^{2} - (x + a)(x + b) = x
^{2}+ (a + b)x + ab - (x + a)(x – b) = x
^{2}+ (a – b)x – ab - (x – a)(x + b) = x
^{2}+ (b – a)x – ab - (x – a)(x – b) = x
^{2}– (a + b)x + ab - (a + b)
^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a – b)
^{3}= a^{3}– b^{3}– 3ab(a – b) - (x + y + z)
^{2}= x^{2}+ y^{2}+ z^{2}+ 2xy + 2yz + 2xz - (x + y – z)
^{2}= x^{2}+ y^{2}+ z^{2}+ 2xy – 2yz – 2xz - (x – y + z)
^{2}= x^{2}+ y^{2}+ z^{2}– 2xy – 2yz + 2xz - (x – y – z)
^{2}= x^{2}+ y^{2}+ z^{2}– 2xy + 2yz – 2xz - x
^{3}+ y^{3}+ z^{3}– 3xyz = (x + y + z)(x^{2}+ y^{2}+ z^{2}– xy – yz − xz) - (x + a)(x + b)(x + c) = x
^{3}+ (a + b + c)x^{2}+ (ab + bc + ca)x + abc - x
^{3}+ y^{3}= (x + y)(x^{2}– xy + y^{2}) - x
^{3}– y^{3}=(x – y)(x^{2}+ xy + y^{2}) - x
^{2}+ y^{2}+ z^{2}− xy – yz – zx = 12[(x − y)^{2}+ (y − z)^{2}+ (z − x)^{2}] - x
^{2}+ y^{2}= 1/2[(x + y)^{2 }+ (x – y)^{2}]

**Formula for Quadratic Equations**

For ax^{2} + bx + c = 0,

(α, β) =[−b±(b2−4ac)/√2ac

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

- If b
^{2}− 4ac > 0, then the quadratic equation has two distinct real roots. - If b
^{2}− 4ac < 0, then the quadratic equation has two imaginary roots. - If b
^{2}− 4ac = 0, then the quadratic equation has two equal real roots.

**Arithmetic Progression Formula**

- n
^{th}term of an arithmetic sequence: a_{n}= a + (n – 1)d - Sum of n terms of an arithmetic sequence: S
_{n}= n/2 2a+(n–1)d - nth term of a geometric sequence: an = a.rn-1
- Sum of n terms of a geometric sequence: Sn = a (1 – rn)/ (1 – r), r≠1
- 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.

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

- Subtraction of x from y.
- One-third of the sum of numbers a and b.
- The number r multiplied by itself.
- One-half of the product of numbers s and t.
- Numbers c and d both squared and subtracted.
- Number 10 added to five times the product of number v and w
- Product of numbers y and x added to 20.
- Sum of numbers a and b and c subtracted from their respective product.

**Answer:**

- y – x
- 1/3 (a +b)
- r2
- ½ (st)
- c2 – d2
- 10 + 5 (vw)
- xy + 20
- abc – (a + b + c)

**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

= 100 + 4n (n + 10)

= 4n2 + 40n + 100

= (2n + 10)2

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**

- Find the roots of the equation y2 – 5y – p (p + 5) = 0, where p is a constant.
- If 1 is a root of the equations az2 + az + 16 = 0 and x2 + x + b = 0, then find the value of zx.
- Find the value of p so that the quadratic equation px(x – 4) + 16 = 0 has two similar roots.
- Find the value of q so that the quadratic equation qx (x – 10) + 100 = 0 has two similar roots.
- Simplify the algebraic expression -2(x – 5) + 4(-3 x + 8)
- Expand and simplify the algebraic x + 2) (x – 2) – (-x – 4)
- For what values of k the equation -x2 + 3kx – 9 = 0 has one real solution?
- 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:

a_{1}x + b_{1 }+ c_{1 }= 0 and a_{2}x + b_{2 }+ c_{2}= 0

When a_{1}, b_{1}, c_{1}, & a_{2}, b_{2}, c_{2} are real numbers where a_{1}^{2}+b_{1}^{2} ≠ 0 & a_{2}^{2 }+ b_{2}^{2} ≠ 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.

## FAQs on Algebra Formula for Class 10

**1. What is Algebraic Equation?**

An equation which is the combination of constants and variables connected by the signs of +,-, / or x is equal to another equation is called an algebraic equation.

A mathematical statement is said to be an algebraic equation if the left-hand side equation is equal to the right-hand side equation. That means values to both sides of equal to sign are equal.

Consider an equation 5 + 2 = 3 + 4 here, you can see both sides have the value 6. Such expressions are called algebraic equations.

**2. What is the Difference Between the Algebraic Equation and Algebraic Expression?**

Students are often confused about the term Algebraic Equation and Algebraic expression. Let us make it more clear.

- An algebraic expression is a single equation with no equality sign, whereas the Algebraic equation is a set of two equations that have equality sign between them.
- An algebraic expression is of form 2x + 3y – x + y.
- An Algebraic equation is of the form 5x + 2 = 0.
- An Algebraic expression can be expanded or simply whereas, an algebraic equation has to be solved, it has a certain value.
- The algebraic expression will be true for any value of x, while the algebraic equation will be true only for some values of x.

**3. Why is it Important to Know all Formulas of Algebra Class 10?**

The gradual up-gradation of difficulty level in mathematical problems with each Class requires efficient and fast calculation procedure. One must be very well versed with the application of Algebra formulas for Class 10 in order to solve complex problems in no time. Be it problems from linear equations or factorization, the in-depth knowledge of Algebraic expressions enables the student to present accurately.

**4. How can we Find the Sum or Difference of Two Numbers Raised to any Power?**

Suppose the two numbers that are to be raised to power are a and b. If they are to be raised to a certain natural number ‘n’, let us know how to find their sum or difference.

**Difference:**

a^{n} – b^{n} = (a – b) (a^{n-1} + a^{n-2}b +….b^{n-2}a + b^{n-1})

**Sum:**

- if ‘n’ is an even number, a
^{n}+ b^{n}= (a + b) (a^{n-1}– a^{n-2}b +….+ b^{n-2}a – b^{n-1}) - if ‘n’ is an odd number a
^{n}+ b^{n}= (a – b) (a^{n-1}– a^{n-2}b +…. – b^{n-2}a + b^{n-1})

**5. In which chapters of CBSE Class 10 mathematics are the algebraic formulas applicable?**

- The Elimination method of algebraic formulas is applicable in solving different pairs of linear equations.
- The Substitution method of algebraic formulas are applicable in solving different pairs of linear equations
- And the Cross-Multiplication method of algebraic formulas is applicable in solving different pairs of linear equations

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