Math Problem Statement
Perform the indicated operations and simplify:
(a) 3(x + 6) + 4(2x − 5)
(b) (x + 3)(4x − 5)
(c) (√a + √b)(√a − √b)
(d) (2x + 3)^2
(e) (x + 2)^3
Factor each expression completely:
(a) 4x^2 − 25
(b) 2x^2 + 5x − 12
(c) x^3 − 3x^2 − 4x + 12
Solution
I have reviewed the image and will now solve the indicated mathematical problems.
Part 1: Perform the operations and simplify
(a)
= (3x + 8x) + (18 - 20) = 11x - 2$$ (b) $$(x + 3)(4x - 5)$$ $$= x(4x - 5) + 3(4x - 5) = 4x^2 - 5x + 12x - 15 = 4x^2 + 7x - 15$$ (c) $$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})$$ $$= (\sqrt{a})^2 - (\sqrt{b})^2 = a - b$$ (d) $$(2x + 3)^2$$ $$= (2x + 3)(2x + 3) = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9$$ (e) $$(x + 2)^3$$ $$= (x + 2)(x + 2)(x + 2) = (x^2 + 4x + 4)(x + 2) = x(x^2 + 4x + 4) + 2(x^2 + 4x + 4) = x^3 + 4x^2 + 4x + 2x^2 + 8x + 8 = x^3 + 6x^2 + 12x + 8$$ ### Part 2: Factor each expression completely (a) $$4x^2 - 25$$ This is a difference of squares: $$= (2x + 5)(2x - 5)$$ (b) $$2x^2 + 5x - 12$$ We find factors of $$-24$$ that add to $$5$$, which are $$8$$ and $$-3$$: $$= 2x^2 + 8x - 3x - 12 = 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4)$$ (c) $$x^3 - 3x^2 - 4x + 12$$ Group the terms: $$= (x^3 - 3x^2) - (4x - 12) = x^2(x - 3) - 4(x - 3) = (x^2 - 4)(x - 3)$$ Factor $$x^2 - 4$$ as a difference of squares: $$= (x + 2)(x - 2)(x - 3)$$ Let me know if you need further clarification, and here are some additional practice questions: 1. Simplify $$5(2x - 4) + 3(3x + 1)$$. 2. Factor $$9x^2 - 16$$. 3. Expand and simplify $$(3x - 2)(x^2 + 4x + 1)$$. 4. Factor $$3x^2 + 14x + 8$$. 5. Solve $$(x + 5)^2 = 49$$. **Tip**: For factoring quadratic expressions, try splitting the middle term to simplify the process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Factoring
Polynomial expansion
Difference of squares
Exponentiation
Formulas
Difference of squares: a^2 - b^2 = (a + b)(a - b)
Binomial expansion: (a + b)^n
Factoring quadratics: ax^2 + bx + c
Theorems
Difference of squares
Quadratic factoring
Suitable Grade Level
Grades 9-10
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