22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!
2 Factoring out a GCF ★★★
Factor each polynomial: a) \(15x + 25\) b) \(8y^2 - 12y\) c) \(9m^3 + 6m^2\).
3 Easy trinomials ★★★
Factor: a) \(x^2 + 7x + 10\) b) \(x^2 + 10x + 21\) c) \(x^2 + 11x + 24\).
4 Trinomials with negative terms ★★★
Factor: a) \(x^2 - x - 12\) b) \(x^2 + 2x - 15\) c) \(x^2 - 9x + 14\).
5 Differences of squares ★★★
Factor: a) \(x^2 - 81\) b) \(4y^2 - 25\) c) \(100 - n^2\).
6 Reading factored equations ★★★
Solve: a) \((x - 4)(x + 9) = 0\) b) \(x(2x - 7) = 0\) c) \((3x + 6)(x - 1) = 0\).
7 True or false? ★★★
Decide whether each statement is true or false and justify. a) \(x^2 + 16 = (x + 4)(x - 4)\) b) \(6x + 9 = 3(2x + 3)\) c) \(x^2 + 10x + 25 = (x + 5)^2\).
8 Factor completely ★★★
Factor \(12x^3 - 18x^2 + 30x\) completely.
9 Grouping practice ★★★
Factor by grouping: a) \(x^3 + 5x^2 + 2x + 10\) b) \(3ab + 6a - 4b - 8\).
10 Larger constants ★★★
Factor: a) \(x^2 - 3x - 54\) b) \(x^2 + x - 56\).
11 The ac method ★★★
Factor: a) \(2x^2 + 7x + 3\) b) \(5x^2 - 17x + 6\).
12 GCF and squares together ★★★
Factor completely: a) \(9x^2 - 49\) b) \(50x^2 - 2\).
13 Perfect square trinomials ★★★
Factor: a) \(x^2 - 12x + 36\) b) \(49x^2 + 28x + 4\) c) \(9y^2 - 24y + 16\).
14 First equations ★★★
Solve by factoring: a) \(x^2 - 8x + 15 = 0\) b) \(x^2 + 3x = 28\).
15 Reading a graph ★★★
The graph shows \(y = x^2 - 2x - 8\).
a) Read the x-intercepts. b) Factor \(x^2 - 2x - 8\) and confirm. c) Where is the y-intercept?
16 Garden dimensions ★★★
A rectangular garden has area \((x^2 + 9x + 14)\) square meters. a) Factor the area to find expressions for the length and the width. b) If \(x = 3\), find the perimeter.
17 A trinomial equation ★★★
Solve \(6x^2 + x - 2 = 0\).
18 Squares and common factors ★★★
Solve: a) \(4x^2 = 25\) b) \(3x^2 = 12x\).
19 Consecutive integers ★★★
The product of two consecutive positive integers is 156. Find them.
20 A launched ball ★★★
A ball thrown upward from a platform has height \(h = -16t^2 + 64t + 80\) feet after \(t\) seconds. When does it hit the ground (\(h = 0\))?
21 Find the error ★★★
A student solves \((x - 2)(x + 3) = 14\) by writing \(x - 2 = 14\) or \(x + 3 = 14\). Explain the mistake and solve correctly.
22 Which values of k? ★★★
Find every integer \(k\) for which \(x^2 + kx + 12\) can be factored into two binomials with integer coefficients.
Test yourself: quick challenge for Grade 9
Speed drill for Grade 9: how many in 60 seconds?
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