1. **Problem Statement:** Revise and practice problems on Algebraic Expressions and Identities covering addition, subtraction, multiplication, and algebraic identities. 2. **Key Concepts:** - Addition and subtraction of algebraic expressions involve combining like terms. - Multiplication of algebraic expressions uses distributive property: $$a(b+c) = ab + ac$$. - Algebraic identities are formulas like $$(a+b)^2 = a^2 + 2ab + b^2$$, $$(a-b)^2 = a^2 - 2ab + b^2$$, and $$a^2 - b^2 = (a-b)(a+b)$$. 3. **Worksheet Questions:** 1. Simplify: $$3x + 5x$$ (1M) 2. Subtract: $$(7y - 3) - (2y + 4)$$ (1M) 3. Multiply: $$2x(3x + 4)$$ (2M) 4. Expand: $$(x + 5)(x - 2)$$ (2M) 5. Factorize: $$x^2 - 9$$ (2M) 6. Simplify using identity: $$(a+b)^2$$ when $$a=2, b=3$$ (2M) 7. Expand and simplify: $$(2x + 3)(x - 4)$$ (5M) 8. Factorize completely: $$x^2 + 5x + 6$$ (5M) **Case-based Study:** Given the expression $$P = (x + 3)^2 - (x - 2)^2$$ 9. Find the value of $$P$$ when $$x=4$$ (1M) 10. Simplify the expression $$P$$ using algebraic identities (1M) 11. Factorize the simplified form of $$P$$ (2M) 4. **Detailed Solutions:** 1. Combine like terms: $$3x + 5x = 8x$$. 2. Subtract terms: $$(7y - 3) - (2y + 4) = 7y - 3 - 2y - 4 = (7y - 2y) + (-3 - 4) = 5y - 7$$. 3. Use distributive property: $$2x(3x + 4) = 2x \times 3x + 2x \times 4 = 6x^2 + 8x$$. 4. Expand: $$(x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10$$. 5. Recognize difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$. 6. Use identity: $$(a+b)^2 = a^2 + 2ab + b^2$$ Substitute $$a=2, b=3$$: $$2^2 + 2 \times 2 \times 3 + 3^2 = 4 + 12 + 9 = 25$$. 7. Expand: $$(2x + 3)(x - 4) = 2x \times x + 2x \times (-4) + 3 \times x + 3 \times (-4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$$. 8. Factorize quadratic: $$x^2 + 5x + 6$$ factors as $$(x + 2)(x + 3)$$. 9. Substitute $$x=4$$ in $$P = (x + 3)^2 - (x - 2)^2$$: $$(4 + 3)^2 - (4 - 2)^2 = 7^2 - 2^2 = 49 - 4 = 45$$. 10. Use identity for difference of squares: $$P = (x + 3)^2 - (x - 2)^2 = [(x + 3) - (x - 2)] \times [(x + 3) + (x - 2)] = (x + 3 - x + 2)(x + 3 + x - 2) = (5)(2x + 1) = 5(2x + 1)$$. 11. Factorized form is $$5(2x + 1)$$.