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{ "quality": "incomplete", "question_count": 15, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Factoring Trinomials", "Solving Quadratic Equations" ], "current_concepts": [ "Factoring Polynomials", "Solving Quadratic Equations", "Quadratic Word Problems" ] } ## Factor polynomials 1 to 10 $$ \begin{aligned} 1.&\quad 2x^2 + 4x - 16 = 2(x^2 + 2x - 8) = 2(x + 4)(x - 2)\ 2.&\quad 5x^2 + 21x + 18 = (5x + 6)(x + 3)\ 3.&\quad 4x^2 - 4x - 63 = (2x - 9)(2x + 7)\ 4.&\quad 6x^2 - 19x + 8 = (3x - 8)(2x - 1)\ 5.&\quad -9x^2 + 31x + 20 = -(9x^2 - 31x - 20) = -(9x + 5)(x - 4)\ 6.&\quad 3x^2 - 6x - 24 = 3(x^2 - 2x - 8) = 3(x - 4)(x + 2)\ 7.&\quad 28x^2 + x - 15 = (7x - 5)(4x + 3)\ 8.&\quad -36x^2 + 30x + 66 = -6(6x^2 - 5x - 11) = -6(6x - 11)(x + 1)\ 9.&\quad -10x^2 - 34x - 12 = -2(5x^2 + 17x + 6) = -2(5x + 2)(x + 3)\ 10.&\quad 6x^2 - 19x - 36 = (2x - 9)(3x + 4) \end{aligned} $$ ## Solve quadratic equations 11 to 14 $$ \begin{aligned} 11.&\quad 4x^2 + 40x + 84 = 0 \implies 4(x^2 + 10x + 21) = 0 \implies 4(x+7)(x+3) = 0 \implies x = -7, -3\ 12.&\quad -3x^2 - 15x + 72 = 0 \implies -3(x^2 + 5x - 24) = 0 \implies -3(x+8)(x-3) = 0 \implies x = -8, 3\ 13.&\quad -15x^2 + 28x - 5 = 0 \implies -(15x^2 - 28x + 5) = 0 \implies -(3x - 5)(5x - 1) = 0 \implies x = \frac{5}{3}, \frac{1}{5}\ 14.&\quad 24x^2 - 47x - 21 = 0 \implies (8x - 21)(3x + 1) = 0 \implies x = \frac{21}{8}, -\frac{1}{3} \end{aligned} $$ ## Solve word problem 15 $$ \begin{aligned} &\text{Let width } = w, \text{ length } = 3w + 2\ &w(3w + 2) = 56 \implies 3w^2 + 2w - 56 = 0 \implies (3w + 14)(w - 4) = 0\ &w = 4 \quad (w > 0)\ &\text{width } = 4\text{ ft}, \text{ length } = 14\text{ ft} \end{aligned} $$ 1. $2(x + 4)(x - 2)$ 2. $(5x + 6)(x + 3)$ 3. $(2x - 9)(2x + 7)$ 4. $(3x - 8)(2x - 1)$ 5. $-(9x + 5)(x - 4)$ 6. $3(x - 4)(x + 2)$ 7. $(7x - 5)(4x + 3)$ 8. $-6(6x - 11)(x + 1)$ 9. $-2(5x + 2)(x + 3)$ 10. $(2x - 9)(3x + 4)$ 11. $x = -7, -3$ 12. $x = -8, 3$ 13. $x = \frac{5}{3}, \frac{1}{5}$ 14. $x = \frac{21}{8}, -\frac{1}{3}$ 15. Width = $4\text{ ft}$, Length = $14\text{ ft}$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Factoring Polynomials" ] }

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Factor polynomials 1 to 10

Solve quadratic equations 11 to 14

Solve word problem 15

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