# Math Blog

### Worksheet on Factorization | Hints | Miscellaneous Factorization

Practice the questions given in the Worksheet on Factorization. Factorization of expressions of the form a^3 ± b^3 1. Factorize: (i) 8x^3 + 27y^3 (ii) 216a^3 + 1 (iii) a^6 + 1 (iv) x^3 + $$\frac{1}{x^{3}}$$ (v) a^3 + 8b^6

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### Factorization of expressions of the Form a^3 + b^3 + c^3 – 3abc

Here we will learn the process of On Factorizations of expressions of the Form a^3 + b^3 + c^3 – 3abc. We have, a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 – bc – ca – ab). [Verify by actual multiplication.] Example: Factorize: x^3 + y^3 – 3xy + 1. Solution: Here

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### Factorization of Expressions of the Form a^3 + b^3 + c^3, a + b + c=0

Here we will learn the process of On Factorization of expressions of the Form a^3 + b^3 + c^3 , where a + b + c = 0. We have, a^3 + b^3 + c^3 = a^3 + b^3 – (-c)^3 = a^3 + b^3 – (a + b)^3, [Since, a + b + c = 0] = a^3 + b^3 – {a^3 + b^3 + 3ab(a + b)} = -3ab(a + b) = -3ab(-c

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### Miscellaneous Problems on Factorization | Application Problems

Here we will solve different types of Miscellaneous Problems on Factorization. 1. Factorize: x(2x + 5) – 3 Solution: Given expression = x(2x + 5) – 3 = 2x^2 + 5x – 3 = 2x^2 + 6x – x – 3, [Since, 2(-3) = - 6 = 6 × (-1), and 6 + (-1) = 5] = 2x(x + 3) – 1(x + 3)

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### Factorization of Expressions of the Form a^3 - b^3 | Solved Examples

Here we will learn the process of Factorization of Expressions of the Form a^3 - b^3. We know that (a - b)^3 = a^3 - b^3 - 3ab(a - b), and so a^3 - b^3 = (a - b)^3 + 3ab(a - b) = (a - b){(a - b)^2 + 3ab} Therefore, a^3 - b^3 = (a - b)(a^2 + ab + b^2) Example: 1. Factorize:

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### Factorization of Expressions of the Form a^3 + b^3 | Solved Examples

Here we will learn the process of Factorization of Expressions of the Form a^3 + b^3. We know that (a + b)^3 = a^3 + b^3 + 3ab(a + b), and so a^3 + b^3 = (a + b)^3 – 3ab(a + b) = (a + b){(a + b)^2 – 3ab} Therefore, a^3 + b^3 = (a + b)(a^2 – ab + b^2). Example: 1. Factorize:

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### Worksheet on Factorization of the Trinomial ax^2 + bx + c | Answers

Practice the questions given in the worksheet on factorization of the trinomial ax^2 + bx + c. 1. Factorization of a perfect-square trinomial. (i) a^2 + 6a + 9 (ii) a^2 + a + $$\frac{1}{4}$$ (iii) 25x^2 – 10x + 1 (iv) 4x^2 – 4xy + y^2 2. Factorization of expressions of the

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### Problems on Factorization of Expressions of the Form x^2 +(a + b)x +ab

Here we will solve different types of Problems on Factorization of Expressions of the Form x^2 + (a + b)x + ab. 1. Factorize: a^2 + 25a - 54 Solution: Here, constant term = -54 = (27) × (-2), and 27 + (-2) = 25 (= coefficient of a). Therefore, a^2 + 25a – 54

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### Factorization of Expressions of the Form ax^2 + bx + c, a ≠ 1|Examples

The below examples show that the method of factorizing ax^2 + bx + c by breaking the middle term involves the following steps. Steps: 1.Take the product of the constant term and the coefficient of x^2, i.e., ac. 2. Break ac into two factors p, q whose sum is b,

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### Factorization of Expressions of the Form x^2 + (a + b)x + ab |Examples

Here we will learn the process of Factorization of Expressions of the Form x^2 + (a + b)x + ab. We know, (x + a)(x + b) = x^2 + (a + b)x + ab. Therefore, x^2 + (a + b)x + ab = (x + a)(x + b). 1. Factorize: a^2 + 7a + 12. Solution: Here, constant term = 12 = 3 × 4, and 3 + 4

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### Factorization of a Perfect-square Trinomial | Solved Examples

Here we will learn the process of Factorization of a Perfect-square Trinomial. A trinomial of the form a^2 ± 2ab + b^2 = (a ± b)^2 = (a ± b)(a ± b) Solved examples on Factorization of a Perfect-square Trinomial 1. Factorize: x^2 + 6x + 9 Solution: Here, given expression

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### Problems on Factorization Using a^2 – b^2 = (a + b)(a – b)

Problems on Factorization using a^2 – b^2 = (a + b)(a – b) Here we will solve different types of Problems on Factorization using a^2 – b^2 = (a + b)(a – b). 1. Factorize: 4a^2 – b^2 + 2a + b Solution: Given expression = 4a^2 – b^2 + 2a + b = (4a^2 – b^2) + 2a + b

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### Problems on Factorization of Expressions of the Form a^2 – b^2

Here we will solve different types of Problems on Factorization of expressions of the form a^2 – b^2. 1. Resolve into factors: 49a^2 – 81b^2 Solution: Given expression = 49a^2 – 81b^2 = (7a)^2 – (9b)^2 = (7a + 9b)(7a – 9b). 2.Factorize: (x + y)^2 – 4(x - y)^2 Solution: Given

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### Problems on Factorization by Grouping of Terms | Find the Factors

Here we will solve different types of Problems on Factorization by grouping of terms. 1. Factorize: a^2 – (b – 5)a – 5b. Solution: Given expression = a^2 – (b – 5)a – 5b = a^2 – ba + 5a – 5b = a(a - b) + 5(a - b) = (a – b)(a + 5). 2. Factorize: a^2 + b^2 + a + b + 2ab

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### Introduction to Factorization | Different of Two Squares | Examples

We will discuss here about the introduction to factorization. The method of expressing a given polynomial as a product of two or more polynomials is called factorization. The polynomials whose product is the given polynomial are called its factors. You are already familiar

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### Problems on Expanding of (a ± b)$$^{3}$$ and its Corollaries |Examples

Here we will solve different types of application problems on expanding of (a ± b)$$^{3}$$ and its corollaries. 1. Expanding the following: (i) (1 + x)$$^{3}$$ (ii) (2a – 3b)$$^{3}$$ (iii) (x + $$\frac{1}{x}$$)$$^{3}$$ Solution: (i) (1 + x)$$^{3}$$ = 1$$^{3}$$ +

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### Worksheet on Application Problems on Expansion of Powers of Binomials

Practice the questions given in the worksheet on application problems on expansion of powers of binomials and trinomials. 1. Use (a ± b)^2 = a^2 ± 2ab + b2 to evaluate the following: (i) (3.001)^2 (ii) (5.99)^2 (iii) 1001 × 999 (iv) 5.63 × 5.63 + 11.26 × 2.37 + 2.37 × 2.37

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### Application Problems on Expansion of Powers of Binomials & Trinomials

Here we will solve different types of application problems on expansion of powers of binomials and trinomials. 1. Use (x ± y)^2 = x^2 ± 2xy + y^2 to evaluate (2.05)^2. Solution: (2.05)^2 = (2 + 0.05)^2 = 2^2 + 2 × 2 × 0.05 + (0.05)^2 = 4 + 0.20 + 0.0025 = 4.2025.

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### Expansion of (x + a)(x + b)(x + c) | Solved Examples | Problems | Hint

We will discuss here about the expansion of (x + a)(x + b)(x + c). (x + a)(x + b)(x + c) = (x + a){(x + b)(x + c)} = (x + a){x$$^{2}$$ + (b + c)x + bc} = x{x$$^{2}$$ + (b + c)x + bc} + a{x$$^{2}$$ + (b + c)x + bc} = x$$^{3}$$ + (b + c)x$$^{2}$$ + bcx + ax$$^{2}$$ + a(b + c)x

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### Simplification of (a + b + c)(a$$^{2}$$+b$$^{2}$$+c$$^{2}$$–ab–bc– ca)

We will discuss here about the expansion of (a + b + c)(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ – ab – bc – ca). (a + b + c)(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ – ab – bc – ca) = a(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ – ab – bc – ca) + b(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ –ab – bc – ca)

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### Simplification of (a ± b)(a^2 ∓ ab + b^2) | Sum or Difference of Cubes

We will discuss here about the expansion of (a ± b)(a$$^{2}$$ ∓ ab + b$$^{2}$$). (a + b)(a$$^{2}$$ - ab + b$$^{2}$$) = a(a$$^{2}$$ - ab + b$$^{2}$$) + b(a$$^{2}$$ - ab + b$$^{2}$$) = a$$^{3}$$ - a$$^{2}$$b + ab$$^{2}$$ + ba$$^{2}$$ - ab$$^{2}$$ + b$$^{3}$$ =

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### Expansion of (a ± b)$$^{3}$$ | Algebraic Expressions and Formulas

We will discuss here about the expansion of (a ± b)$$^{3}$$. (a + b)$$^{3}$$ = (a + b) ∙ (a + b)$$^{2}$$ = (a + b)(a$$^{2}$$ + 2ab + b$$^{2}$$) = a(a$$^{2}$$ + 2ab + b$$^{2}$$) + b(a$$^{2}$$ + 2ab + b$$^{2}$$)=a$$^{3}$$+2a$$^{2}$$b+ab$$^{2}$$+ba$$^{2}$$+2ab$$^{2}$$+b$$^{3}$$

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### Express a^2 + b^2 + c^2 – ab – bc – ca as Sum of Squares

Here we will express a^2 + b^2 + c^2 – ab – bc – ca as sum of squares. If a, b, c are real numbers then (a – b)^2, (b – c)^2 and (c – a)^2 are positive as square of every real number is positive. So, a^2 + b^2 + c^2 – ab – bc – ca is always positive.

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### Worksheet on Simplification of (a + b)(a – b) | Hint | Answers

Practice the questions given in the worksheet on simplification of (a + b)(a – b). 1. Simplify by applying standard formula. (i) (5x – 9)(5x + 9) (ii) (2x + 3y)(2x – 3y) (iii) (a + b – c)(a – b + c) (iv) (x + y – 3)(x + y + 3) (v) (1 + a)(1 – a)(1 + a^2)

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### Worksheet on Completing Square |Find the Missing Term| Perfect Squares

Practice the questions given in the worksheet on completing square. Write the following as a perfect square. (i) 4X^2 + 4X + 1 (ii) 9a^2 – 12ab + 4b^2 (iii) 1 + 6/a + 9/a^2 2. Indicate the perfect squares among the following. Express each of the perfect squares as the square

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### Completing a Square | Solved Examples on Completing a Square

Here we will learn how to completing a square.Problems on completing a square 1. What should be added to the polynomial 4m^2 + 8m so that it becomes perfect square? Solution: 4m^2 + 8m = (2m)^2 + 2 ∙ (2m) ∙ 2

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### Worksheet on Expansion of (x ± a)(x ± b) | Find the Product | Answers

Practice the questions given in the worksheet on expansion of (x ± a)(x ± b). 1. (i) Find the product using standard formula. (i) (x + 2)(x + 5) (ii) (a – 4)(a – 7) (iii) (x + 1)(x – 8) (iv) (a – 3)(a + 2) (v) (3x + 1)(3x + 2) (vi) (4x – y)(4x + 2y) 2. Find the product.

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### Worksheet on Expanding of (a ± b ± c)^2 and its Corollaries | Answers

Practice the questions given in the worksheet on expanding of (a ± b ± c)^2 and its corollaries. 1. Expand the squares of the following trinomials. (i) a + 2b + 3c (ii) 2x + 3y + 4z (iii) x + 2y – 3z (iv) 3a – 4b – c (v) 1 – x - $$\frac{1}{x}$$ (vi) 1 – a – a^2. 2. Simplify:

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### Worksheet on Expansion of (a ± b)^2 and its Corollaries | Answers

Practice the questions given in the worksheet on expansion of (a ± b)^2 and its corollaries. 1. Expand the squares of the following: (i) 4x + y (ii) 5a + 3b (iii) 2x + $$\frac{1}{x}$$ 2. Expand the following: (i) (x – 2y)^2 (ii) (3y – 2z)^2 (iii) (3x - $$\frac{1}{3x}$$)^2

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### Simplification of (a + b)(a – b) | | Solved Examples on Simplification

We will discuss here about the Simplification of (a + b)(a – b). (a + b)(a – b) = a(a – b) + b(a – b) = a$$^{2}$$ - ab + ba - b$$^{2}$$ = a$$^{2}$$ - b$$^{2}$$ Thus, we have (a + b)(a - b) = a$$^{2}$$ - b$$^{2}$$ Solved Examples on Simplification of (a + b)(a – b) 1.

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### Expansion of (x ± a)(x ± b) | Special Identities | Expanding Binomials

We will discuss here about the expansion of (x ± a)(x ± b) (x + a)(x + b) = x(x + b) + a (x + b) = x^2 + xb + ax + ab = x^2 + (b + a)x + ab (x - a)(x - b) = x(x - b) - a (x - b) = x^2 - xb - ax + ab = x^2 - (b + a)x + ab (x + a)(x - b) = x(x - b) + a (x - b) = x^2 - xb

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### Expansion of (a ± b ± c)^2 | Square of a Trinomial | Algebra Formulas

We will discuss here about the expansion of (a ± b ± c)^2. (a + b + c)^2 = {a + (b + c)}^2 = a^2 + 2a(b + c) + (b + c)^2 = a^2 + 2ab + 2ac + b^2 + 2bc + c^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) = sum of squares of a, b, c + 2(sum of the products of a, b, c taking two at a ti

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### A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles

Here we will prove that a rhombus is a parallelogram whose diagonals meet at right angles. Given: PQRS is a rhombus. So, by definition, PQ = QR = RD = SP. Its diagonals PR and QS intersect at O. To prove: (i) PQRS is a parallelogram. (ii) ∠POQ = ∠QOR = ∠ROS = ∠SOP = 90°.

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### Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Here we will discuss about one of the important geometrical property of parallelogram. A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR. To prove: PQRS is a parallelogram.

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### A Quadrilateral is a Parallelogram if its Diagonals Bisect each Other

Here we will discuss about a quadrilateral is a parallelogram if its diagonals bisect each other. Given: PQRS is a quadrilateral whose diagonals PR and QS bisect each other at O, i.e., OP = OR and OQ = OS. To prove: PQRS is a parallelogram. Proof: In ∆OPQ and ∆ORS, OP = OR

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### Diagonals of a Parallelogram Bisect each Other | Diagonals Bisect each

Here we will discuss about the diagonals of a parallelogram bisect each other. In a parallelogram, diagonals bisect each other and each diagonal bisects the parallelogram into two congruent triangles. Given: PQRS is a parallelogram in which PQ ∥ SR and PS ∥ QR. Its diagonals

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### Expansion of (a ± b)^2 | Power of the Trinomial | Algebraic Expression

A binomial is an algebraic expression which has exactly two terms, for example, a ± b. Its power is indicated by a superscript. For example, (a ± b)2 is a power of the binomial a ± b, the index being 2. A trinomial is an algebraic expression which has exactly three terms

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### Opposite Angles of a Parallelogram are Equal | Related Solved Examples

Here we will discuss about the opposite angles of a parallelogram are equal. In a parallelogram, each pair of opposite angles are equal. Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS To prove: ∠P = ∠R and ∠Q = ∠S Construction: Join PR and QS. Proof: Statement:

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### Opposite Sides of a Parallelogram are Equal | Solved Examples

Here we will discuss about the opposite sides of a parallelogram are equal in length. In a parallelogram, each pair of opposite sides are of equal length. Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS. To prove: PQ = SR and PS = QR. Construction: Join PR

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### Concept of Parallelogram |Quadrilateral| Rectangle| Rhombus| Trapezium

Here we will discuss about the concept of parallelogram. Quadrilateral: A rectilinear figure enclosed by four line segments is called a quadrilateral. In the adjoining figures, we have two quadrilaterals PQRS, each enclosed by four line segments PQ, QR, RS and SP which

### Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area

Here we will prove that if each diagonal of a quadrilateral divides it in two triangles of equal area then prove that the quadrilateral is a parallelogram. Solution: Given: PQRS is a quadrilateral whose diagonals PR and QS cut at O such that ar(∆PQR) = ar(∆PSR), and

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### The Area of a Rhombus is Equal to Half the Product of its Diagonals

Here we will prove that the area of a rhombus is equal to half the product of its diagonals. Solution: Given: PQRS is a rhombus whose diagonals are PR and QS. The diagonals intersect at O. To prove: ar(rhombus PQRS) = 1/2 ×PR × QS. Statement ar(∆RSQ) = 1/2 ×Base × Altitude

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### Area of the Triangle formed by Joining the Middle Points of the Sides

Here we will prove that the area of the triangle formed by joining the middle points of the sides of a triangle is equal to one-fourth area of the given triangle. Solution: Given: X, Y and Z are the middle points of sides QR, RP and PQ respectively of the triangle PQR.

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### Problems on Finding Area of Triangle and Parallelogram | With Diagram

Here we will learn how to solve different types of problems on finding area of triangle and parallelogram. 1. In the figure, XQ ∥ SY, PS ∥ QR, XS ⊥ SY, QY ⊥ SY and QY = 3 cm. Find the areas of ∆MSR and parallelogram PQRS. Solution: ar(∆MSR) = 1/2 × ar(rectangle of SR

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### Triangles with Equal Areas on the Same Base have Equal Corresponding..

Here we will prove that triangles with equal areas on the same base have equal corresponding altitudes (or are between the same parallels). Given: PQR and SQR are two triangles on the same base QR, and ar(∆PQR) = ar(∆SQC). Also, PN and SM are their corresponding altitudes.

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### Triangles on the Same Base & between Same Parallels are Equal in Area

Here we will prove that triangles on the same base and between the same parallels are equal in area. Given: PQR and SQR are two triangles on the same base QR and are between the same parallel lines QR and MN, i.e., P and S are on MN. To prove: ar(∆PQR) = ar(∆SQR)

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### Area of a Triangle is Half that of a Parallelogram on the Same Base

Here we will prove that the area of a triangle is half that of a parallelogram on the same base and between the same parallels. Given: PQRS is a parallelogram and PQM is a triangle with the same base PQ, and are between the same parallel lines PQ and SR.

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### Area of a Parallelogram is Equal to that of a Rectangle Between ......

Here we will prove that the area of a parallelogram is equal to that of a rectangle on the same base and of the same altitude, that is between the same parallel lines. Given: PQRS is a parallelogram and PQ MN is a rectangle on the same base PQ and between the same parallel

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### Parallelogram on the Same Base and Between the Same Parallel Lines

Here we will prove that parallelogram on the same base and between the same parallel lines are equal in area. Given: PQRS and PQMN are two parallelograms on the same base PQ and between same parallel lines PQ and SM. To prove: ar(parallelogram PQRS) = ar(parallelogram PQMN).

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