Factoring Quadratics 4C81A6

1. **Problem:** Factor $x^2 + 36x$ completely. 2. **Formula:** For factoring expressions like $ax^2 + bx$, factor out the greatest common factor (GCF). 3. **Work:** $$x^2 + 36x = x(x + 36)$$ 4. **Explanation:** We factor out $x$ because it is common in both terms. --- 1. **Problem:** Factor $2x^2 + 7x + 6$ completely. 2. **Formula:** For quadratic $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. 3. **Work:** $$a=2, b=7, c=6$$ Find two numbers that multiply to $2 \times 6 = 12$ and add to $7$: these are $3$ and $4$. Rewrite middle term: $$2x^2 + 3x + 4x + 6$$ Group: $$(2x^2 + 3x) + (4x + 6)$$ Factor each group: $$x(2x + 3) + 2(2x + 3)$$ Factor out common binomial: $$(x + 2)(2x + 3)$$ 4. **Explanation:** We split the middle term to factor by grouping. --- 1. **Problem:** Factor $p^2 + 5p + 4$ completely. 2. **Formula:** Find two numbers that multiply to $4$ and add to $5$. 3. **Work:** Numbers are $4$ and $1$. $$p^2 + 4p + p + 4 = (p^2 + 4p) + (p + 4) = p(p + 4) + 1(p + 4) = (p + 1)(p + 4)$$ 4. **Explanation:** Factored by grouping after splitting the middle term. --- 1. **Problem:** Factor $125x^2 - 25$ completely. 2. **Formula:** Recognize difference of squares: $a^2 - b^2 = (a - b)(a + b)$. 3. **Work:** $$125x^2 - 25 = 25(5x^2 - 1) = 25((\sqrt{5}x)^2 - 1^2) = 25(\sqrt{5}x - 1)(\sqrt{5}x + 1)$$ 4. **Explanation:** Factored out GCF $25$ and then used difference of squares. --- 1. **Problem:** Factor $x^2 + 2x - 24$ completely. 2. **Formula:** Find two numbers that multiply to $-24$ and add to $2$. 3. **Work:** Numbers are $6$ and $-4$. $$x^2 + 6x - 4x - 24 = (x^2 + 6x) - (4x + 24) = x(x + 6) - 4(x + 6) = (x - 4)(x + 6)$$ 4. **Explanation:** Factored by grouping after splitting the middle term. --- 1. **Problem:** Factor $27x^2 - 3$ completely. 2. **Formula:** Factor out GCF and recognize difference of squares if possible. 3. **Work:** $$27x^2 - 3 = 3(9x^2 - 1) = 3((3x)^2 - 1^2) = 3(3x - 1)(3x + 1)$$ 4. **Explanation:** Factored out $3$ and then used difference of squares. --- 1. **Problem:** Factor $7x^2 - 45x - 28$ completely. 2. **Formula:** Find two numbers that multiply to $7 \times (-28) = -196$ and add to $-45$. 3. **Work:** Numbers are $-49$ and $4$. Rewrite middle term: $$7x^2 - 49x + 4x - 28$$ Group: $$(7x^2 - 49x) + (4x - 28)$$ Factor each group: $$7x(x - 7) + 4(x - 7)$$ Factor out common binomial: $$(7x + 4)(x - 7)$$ 4. **Explanation:** Used factoring by grouping after splitting the middle term. --- 1. **Problem:** Factor $2x^2 + 4x - 6$ completely. 2. **Formula:** Factor out GCF first. 3. **Work:** $$2x^2 + 4x - 6 = 2(x^2 + 2x - 3)$$ Factor inside parentheses: Find two numbers that multiply to $-3$ and add to $2$: $3$ and $-1$. $$2(x^2 + 3x - x - 3) = 2((x^2 + 3x) - (x + 3)) = 2(x(x + 3) - 1(x + 3)) = 2(x - 1)(x + 3)$$ 4. **Explanation:** Factored out GCF and then factored quadratic inside. --- 1. **Problem:** Factor $2k^2 + 6k + 4$ completely. 2. **Formula:** Factor out GCF first. 3. **Work:** $$2k^2 + 6k + 4 = 2(k^2 + 3k + 2)$$ Factor inside parentheses: Find two numbers that multiply to $2$ and add to $3$: $1$ and $2$. $$2(k^2 + k + 2k + 2) = 2((k^2 + k) + (2k + 2)) = 2(k(k + 1) + 2(k + 1)) = 2(k + 2)(k + 1)$$ 4. **Explanation:** Factored out GCF and then factored quadratic inside. --- 1. **Problem:** Factor $x^2 + 14x + 40$ completely. 2. **Formula:** Find two numbers that multiply to $40$ and add to $14$. 3. **Work:** Numbers are $10$ and $4$. $$x^2 + 10x + 4x + 40 = (x^2 + 10x) + (4x + 40) = x(x + 10) + 4(x + 10) = (x + 4)(x + 10)$$ 4. **Explanation:** Factored by grouping after splitting the middle term. **Final answers:** 1. $x(x + 36)$ 2. $(x + 2)(2x + 3)$ 3. $(p + 1)(p + 4)$ 4. $25(\sqrt{5}x - 1)(\sqrt{5}x + 1)$ 5. $(x - 4)(x + 6)$ 6. $3(3x - 1)(3x + 1)$ 7. $(7x + 4)(x - 7)$ 8. $2(x - 1)(x + 3)$ 9. $2(k + 2)(k + 1)$ 10. $(x + 4)(x + 10)$