Factoring Expressions 11A17C

1. **State the problem:** We are asked to factor or simplify the following expressions: (a) $1 - 4b^2$ (b) $49 - 42y + 9y^2$ (c) $2x^2 + 5x + 6$ --- 2. **Factoring (a) $1 - 4b^2$:** This is a difference of squares since $1 = 1^2$ and $4b^2 = (2b)^2$. Formula: $$a^2 - b^2 = (a - b)(a + b)$$ Apply the formula: $$1 - 4b^2 = (1 - 2b)(1 + 2b)$$ --- 3. **Factoring (b) $49 - 42y + 9y^2$:** Rewrite in standard quadratic form: $$9y^2 - 42y + 49$$ Check if it is a perfect square trinomial: $$9y^2 = (3y)^2, \, 49 = 7^2$$ Check middle term: $$2 \times 3y \times 7 = 42y$$ Since middle term is $-42y$, the trinomial is: $$(3y - 7)^2$$ --- 4. **Factoring (c) $2x^2 + 5x + 6$:** Try to factor into form: $$(ax + b)(cx + d)$$ Multiply $a \times c = 2$, possible pairs: (2,1) or (1,2) Multiply $b \times d = 6$, possible pairs: (1,6), (2,3), (3,2), (6,1) Try combinations to get middle term $5x$: Try $(2x + 3)(x + 2)$: $$2x \times x = 2x^2$$ $$2x \times 2 = 4x$$ $$3 \times x = 3x$$ $$3 \times 2 = 6$$ Sum middle terms: $4x + 3x = 7x$ (too big) Try $(2x + 1)(x + 6)$: Middle term sum: $2x \times 6 + 1 \times x = 12x + x = 13x$ (too big) Try $(2x + 2)(x + 3)$: Middle term sum: $2x \times 3 + 2 \times x = 6x + 2x = 8x$ (too big) Try $(2x + 6)(x + 1)$: Middle term sum: $2x \times 1 + 6 \times x = 2x + 6x = 8x$ (too big) Try $(x + 3)(2x + 2)$: Same as above, 8x Try $(x + 2)(2x + 3)$: Middle term sum: $x \times 3 + 2 \times 2x = 3x + 4x = 7x$ (too big) No integer factorization possible. Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{25 - 48}}{4} = \frac{-5 \pm \sqrt{-23}}{4}$$ Since discriminant is negative, no real factors. --- **Final answers:** (a) $1 - 4b^2 = (1 - 2b)(1 + 2b)$ (b) $49 - 42y + 9y^2 = (3y - 7)^2$ (c) $2x^2 + 5x + 6$ is not factorable over the real numbers.