Question: 3. Expand each of the following to standard form and state the y-intercept. [6 marks]\na) y = (x − 3)(x + 4)\nb) y = -2(x + 2)² + 7\n3. Factor each of the following. [8 marks]\na) x² + 8x + 15\nb) x² - 3x - 18\nc) x² - 49\nd) x² + 20x\ne) 3x² + 21x + 36\nf) -6x² + 6x + 36\n6. For the following quadratic relation, y = -2x² + 12x - 10 determine the: y-intercept [1 mark] x-intercepts [4 marks] axis of symmetry [1 mark] vertex [2 marks] Show all of your work, and label the coordinates of all points on the sketch below.

1. **Problem Statement:** Expand and factor given quadratic expressions and analyze the quadratic relation $$y = -2x^2 + 12x - 10$$ to find intercepts, axis of symmetry, and vertex. --- ### Part 3a: Expand $$y = (x - 3)(x + 4)$$ and find y-intercept 2. Use distributive property (FOIL): $$y = x(x + 4) - 3(x + 4) = x^2 + 4x - 3x - 12$$ 3. Simplify like terms: $$y = x^2 + (4x - 3x) - 12 = x^2 + x - 12$$ 4. Find y-intercept by substituting $$x=0$$: $$y = 0^2 + 0 - 12 = -12$$ **Answer:** Standard form $$y = x^2 + x - 12$$, y-intercept at $$(0, -12)$$. --- ### Part 3b: Expand $$y = -2(x + 2)^2 + 7$$ and find y-intercept 5. Expand square: $$y = -2(x + 2)(x + 2) + 7$$ $$y = -2(x^2 + 2x + 2x + 4) + 7$$ 6. Simplify inside parentheses: $$y = -2(x^2 + 4x + 4) + 7$$ 7. Distribute $$-2$$: $$y = -2x^2 - 8x - 8 + 7$$ 8. Simplify constants: $$y = -2x^2 - 8x - 1$$ 9. Find y-intercept by substituting $$x=0$$: $$y = -2(0)^2 - 8(0) - 1 = -1$$ **Answer:** Standard form $$y = -2x^2 - 8x - 1$$, y-intercept at $$(0, -1)$$. --- ### Part 3: Factor each expression 10. a) $$x^2 + 8x + 15$$ Find two numbers that multiply to 15 and add to 8: 3 and 5. $$x^2 + 8x + 15 = (x + 3)(x + 5)$$ 11. b) $$x^2 - 3x - 18$$ Find two numbers that multiply to -18 and add to -3: -6 and 3. $$x^2 - 3x - 18 = (x - 6)(x + 3)$$ 12. c) $$x^2 - 49$$ Difference of squares: $$x^2 - 7^2 = (x - 7)(x + 7)$$ 13. d) $$x^2 + 20x$$ Factor out common factor $$x$$: $$x(x + 20)$$ 14. e) $$3x^2 + 21x + 36$$ Factor out 3: $$3(x^2 + 7x + 12)$$ Factor quadratic inside: $$3(x + 3)(x + 4)$$ 15. f) $$-6x^2 + 6x + 36$$ Factor out $$-6$$: $$-6(x^2 - x - 6)$$ Factor quadratic inside: $$-6(x - 3)(x + 2)$$ --- ### Part 6: Analyze $$y = -2x^2 + 12x - 10$$ 16. **Find y-intercept:** Substitute $$x=0$$: $$y = -2(0)^2 + 12(0) - 10 = -10$$ 17. **Find x-intercepts:** Set $$y=0$$: $$0 = -2x^2 + 12x - 10$$ Divide both sides by $$-2$$: $$0 = \cancel{-2}x^2 \cancel{/\cancel{-2}} - \cancel{12}x \cancel{/\cancel{-2}} + \cancel{10} \cancel{/\cancel{-2}}$$ $$0 = x^2 - 6x + 5$$ 18. Factor quadratic: $$0 = (x - 5)(x - 1)$$ 19. Solve for $$x$$: $$x = 5$$ or $$x = 1$$ 20. **Axis of symmetry:** Formula $$x = -\frac{b}{2a}$$ for $$ax^2 + bx + c$$ Here $$a = -2$$, $$b = 12$$ $$x = -\frac{12}{2(-2)} = -\frac{12}{-4} = 3$$ 21. **Vertex:** Substitute $$x=3$$ into equation: $$y = -2(3)^2 + 12(3) - 10 = -2(9) + 36 - 10 = -18 + 36 - 10 = 8$$ Vertex at $$(3, 8)$$ --- ### Summary: - Part 3a: $$y = x^2 + x - 12$$, y-intercept $$(0, -12)$$ - Part 3b: $$y = -2x^2 - 8x - 1$$, y-intercept $$(0, -1)$$ - Factoring results: a) $$(x + 3)(x + 5)$$ b) $$(x - 6)(x + 3)$$ c) $$(x - 7)(x + 7)$$ d) $$x(x + 20)$$ e) $$3(x + 3)(x + 4)$$ f) $$-6(x - 3)(x + 2)$$ - Quadratic relation $$y = -2x^2 + 12x - 10$$: - y-intercept: $$(0, -10)$$ - x-intercepts: $$(1, 0)$$ and $$(5, 0)$$ - Axis of symmetry: $$x = 3$$ - Vertex: $$(3, 8)$$