10 Printable Math Worksheets - Free Printable

$10 Printable Math Worksheets$

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To solve the problem of factoring the quadratic expressions, we will use various methods such as:

1. Factoring by Grouping
2. Difference of Squares
3. Perfect Square Trinomials
4. Trial and Error for General Quadratics

Let's go through each section step by step.

---

Section A: Factoring Quadratic Expressions

#### 1) $ 2x^2 + x - 1 $
We look for two numbers that multiply to $ 2 \cdot (-1) = -2 $ and add to $ 1 $. These numbers are $ 2 $ and $ -1 $.
\[
2x^2 + x - 1 = 2x^2 + 2x - x - 1 = 2x(x + 1) - 1(x + 1) = (2x - 1)(x + 1)
\]
Answer: $ (2x - 1)(x + 1) $

#### 2) $ 2x^2 + x - 3 $
We look for two numbers that multiply to $ 2 \cdot (-3) = -6 $ and add to $ 1 $. These numbers are $ 3 $ and $ -2 $.
\[
2x^2 + x - 3 = 2x^2 + 3x - 2x - 3 = x(2x + 3) - 1(2x + 3) = (x - 1)(2x + 3)
\]
Answer: $ (x - 1)(2x + 3) $

#### 3) $ 2x^2 + 9x - 5 $
We look for two numbers that multiply to $ 2 \cdot (-5) = -10 $ and add to $ 9 $. These numbers are $ 10 $ and $ -1 $.
\[
2x^2 + 9x - 5 = 2x^2 + 10x - x - 5 = 2x(x + 5) - 1(x + 5) = (2x - 1)(x + 5)
\]
Answer: $ (2x - 1)(x + 5) $

#### 4) $ 2x^2 - 3x - 2 $
We look for two numbers that multiply to $ 2 \cdot (-2) = -4 $ and add to $ -3 $. These numbers are $ -4 $ and $ 1 $.
\[
2x^2 - 3x - 2 = 2x^2 - 4x + x - 2 = 2x(x - 2) + 1(x - 2) = (2x + 1)(x - 2)
\]
Answer: $ (2x + 1)(x - 2) $

#### 5) $ 2x^2 - 13x - 24 $
We look for two numbers that multiply to $ 2 \cdot (-24) = -48 $ and add to $ -13 $. These numbers are $ -16 $ and $ 3 $.
\[
2x^2 - 13x - 24 = 2x^2 - 16x + 3x - 24 = 2x(x - 8) + 3(x - 8) = (2x + 3)(x - 8)
\]
Answer: $ (2x + 3)(x - 8) $

#### 6) $ 3x^2 - 14x - 5 $
We look for two numbers that multiply to $ 3 \cdot (-5) = -15 $ and add to $ -14 $. These numbers are $ -15 $ and $ 1 $.
\[
3x^2 - 14x - 5 = 3x^2 - 15x + x - 5 = 3x(x - 5) + 1(x - 5) = (3x + 1)(x - 5)
\]
Answer: $ (3x + 1)(x - 5) $

#### 7) $ 3x^2 - 8x - 11 $
We look for two numbers that multiply to $ 3 \cdot (-11) = -33 $ and add to $ -8 $. These numbers are $ -11 $ and $ 3 $.
\[
3x^2 - 8x - 11 = 3x^2 - 11x + 3x - 11 = x(3x - 11) + 1(3x - 11) = (x + 1)(3x - 11)
\]
Answer: $ (x + 1)(3x - 11) $

#### 8) $ 2x^2 - 14x + 12 $
First, factor out the greatest common factor (GCF), which is $ 2 $:
\[
2x^2 - 14x + 12 = 2(x^2 - 7x + 6)
\]
Now, factor $ x^2 - 7x + 6 $. We look for two numbers that multiply to $ 6 $ and add to $ -7 $. These numbers are $ -6 $ and $ -1 $.
\[
x^2 - 7x + 6 = (x - 6)(x - 1)
\]
So,
\[
2x^2 - 14x + 12 = 2(x - 6)(x - 1)
\]
Answer: $ 2(x - 6)(x - 1) $

#### 9) $ 3x^2 - 21x + 36 $
First, factor out the GCF, which is $ 3 $:
\[
3x^2 - 21x + 36 = 3(x^2 - 7x + 12)
\]
Now, factor $ x^2 - 7x + 12 $. We look for two numbers that multiply to $ 12 $ and add to $ -7 $. These numbers are $ -4 $ and $ -3 $.
\[
x^2 - 7x + 12 = (x - 4)(x - 3)
\]
So,
\[
3x^2 - 21x + 36 = 3(x - 4)(x - 3)
\]
Answer: $ 3(x - 4)(x - 3) $

#### 10) $ 5x^2 - 41x + 8 $
We look for two numbers that multiply to $ 5 \cdot 8 = 40 $ and add to $ -41 $. These numbers are $ -40 $ and $ -1 $.
\[
5x^2 - 41x + 8 = 5x^2 - 40x - x + 8 = 5x(x - 8) - 1(x - 8) = (5x - 1)(x - 8)
\]
Answer: $ (5x - 1)(x - 8) $

#### 11) $ 3x^2 - 2x - 21 $
We look for two numbers that multiply to $ 3 \cdot (-21) = -63 $ and add to $ -2 $. These numbers are $ -9 $ and $ 7 $.
\[
3x^2 - 2x - 21 = 3x^2 - 9x + 7x - 21 = 3x(x - 3) + 7(x - 3) = (3x + 7)(x - 3)
\]
Answer: $ (3x + 7)(x - 3) $

#### 12) $ 2x^2 + 2x - 12 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 2x - 12 = 2(x^2 + x - 6)
\]
Now, factor $ x^2 + x - 6 $. We look for two numbers that multiply to $ -6 $ and add to $ 1 $. These numbers are $ 3 $ and $ -2 $.
\[
x^2 + x - 6 = (x + 3)(x - 2)
\]
So,
\[
2x^2 + 2x - 12 = 2(x + 3)(x - 2)
\]
Answer: $ 2(x + 3)(x - 2) $

#### 13) $ 2x^2 - 11x + 15 $
We look for two numbers that multiply to $ 2 \cdot 15 = 30 $ and add to $ -11 $. These numbers are $ -6 $ and $ -5 $.
\[
2x^2 - 11x + 15 = 2x^2 - 6x - 5x + 15 = 2x(x - 3) - 5(x - 3) = (2x - 5)(x - 3)
\]
Answer: $ (2x - 5)(x - 3) $

#### 14) $ 3x^2 - 34x - 24 $
We look for two numbers that multiply to $ 3 \cdot (-24) = -72 $ and add to $ -34 $. These numbers are $ -36 $ and $ 2 $.
\[
3x^2 - 34x - 24 = 3x^2 - 36x + 2x - 24 = 3x(x - 12) + 2(x - 12) = (3x + 2)(x - 12)
\]
Answer: $ (3x + 2)(x - 12) $

#### 15) $ 5x^2 - 27x + 10 $
We look for two numbers that multiply to $ 5 \cdot 10 = 50 $ and add to $ -27 $. These numbers are $ -25 $ and $ -2 $.
\[
5x^2 - 27x + 10 = 5x^2 - 25x - 2x + 10 = 5x(x - 5) - 2(x - 5) = (5x - 2)(x - 5)
\]
Answer: $ (5x - 2)(x - 5) $

---

Section B: Factoring Quadratic Expressions

#### 1) $ x^2 + 7x - 30 $
We look for two numbers that multiply to $ -30 $ and add to $ 7 $. These numbers are $ 10 $ and $ -3 $.
\[
x^2 + 7x - 30 = (x + 10)(x - 3)
\]
Answer: $ (x + 10)(x - 3) $

#### 2) $ x^2 + 9x + 20 $
We look for two numbers that multiply to $ 20 $ and add to $ 9 $. These numbers are $ 5 $ and $ 4 $.
\[
x^2 + 9x + 20 = (x + 5)(x + 4)
\]
Answer: $ (x + 5)(x + 4) $

#### 3) $ x^2 + 8x - 9 $
We look for two numbers that multiply to $ -9 $ and add to $ 8 $. These numbers are $ 9 $ and $ -1 $.
\[
x^2 + 8x - 9 = (x + 9)(x - 1)
\]
Answer: $ (x + 9)(x - 1) $

#### 4) $ x^2 - 18x + 80 $
We look for two numbers that multiply to $ 80 $ and add to $ -18 $. These numbers are $ -10 $ and $ -8 $.
\[
x^2 - 18x + 80 = (x - 10)(x - 8)
\]
Answer: $ (x - 10)(x - 8) $

#### 5) $ x^2 - 11x + 28 $
We look for two numbers that multiply to $ 28 $ and add to $ -11 $. These numbers are $ -7 $ and $ -4 $.
\[
x^2 - 11x + 28 = (x - 7)(x - 4)
\]
Answer: $ (x - 7)(x - 4) $

#### 6) $ x^2 + 6x - 72 $
We look for two numbers that multiply to $ -72 $ and add to $ 6 $. These numbers are $ 12 $ and $ -6 $.
\[
x^2 + 6x - 72 = (x + 12)(x - 6)
\]
Answer: $ (x + 12)(x - 6) $

#### 7) $ x^2 - 9x - 22 $
We look for two numbers that multiply to $ -22 $ and add to $ -9 $. These numbers are $ -11 $ and $ 2 $.
\[
x^2 - 9x - 22 = (x - 11)(x + 2)
\]
Answer: $ (x - 11)(x + 2) $

#### 8) $ x^2 - x - 12 $
We look for two numbers that multiply to $ -12 $ and add to $ -1 $. These numbers are $ -4 $ and $ 3 $.
\[
x^2 - x - 12 = (x - 4)(x + 3)
\]
Answer: $ (x - 4)(x + 3) $

#### 9) $ x^2 + 3x - 108 $
We look for two numbers that multiply to $ -108 $ and add to $ 3 $. These numbers are $ 12 $ and $ -9 $.
\[
x^2 + 3x - 108 = (x + 12)(x - 9)
\]
Answer: $ (x + 12)(x - 9) $

#### 10) $ x^2 - 17x + 72 $
We look for two numbers that multiply to $ 72 $ and add to $ -17 $. These numbers are $ -8 $ and $ -9 $.
\[
x^2 - 17x + 72 = (x - 8)(x - 9)
\]
Answer: $ (x - 8)(x - 9) $

#### 11) $ x^2 - x - 42 $
We look for two numbers that multiply to $ -42 $ and add to $ -1 $. These numbers are $ -7 $ and $ 6 $.
\[
x^2 - x - 42 = (x - 7)(x + 6)
\]
Answer: $ (x - 7)(x + 6) $

#### 12) $ x^2 - 15x + 56 $
We look for two numbers that multiply to $ 56 $ and add to $ -15 $. These numbers are $ -7 $ and $ -8 $.
\[
x^2 - 15x + 56 = (x - 7)(x - 8)
\]
Answer: $ (x - 7)(x - 8) $

---

Section C: Factoring Quadratic Expressions

#### 1) $ 2x^2 + 3x + 1 $
We look for two numbers that multiply to $ 2 \cdot 1 = 2 $ and add to $ 3 $. These numbers are $ 2 $ and $ 1 $.
\[
2x^2 + 3x + 1 = 2x^2 + 2x + x + 1 = 2x(x + 1) + 1(x + 1) = (2x + 1)(x + 1)
\]
Answer: $ (2x + 1)(x + 1) $

#### 2) $ 2x^2 + 5x + 2 $
We look for two numbers that multiply to $ 2 \cdot 2 = 4 $ and add to $ 5 $. These numbers are $ 4 $ and $ 1 $.
\[
2x^2 + 5x + 2 = 2x^2 + 4x + x + 2 = 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2)
\]
Answer: $ (2x + 1)(x + 2) $

#### 3) $ 2x^2 + 7x + 3 $
We look for two numbers that multiply to $ 2 \cdot 3 = 6 $ and add to $ 7 $. These numbers are $ 6 $ and $ 1 $.
\[
2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
\]
Answer: $ (2x + 1)(x + 3) $

#### 4) $ 2x^2 + 7x + 5 $
We look for two numbers that multiply to $ 2 \cdot 5 = 10 $ and add to $ 7 $. These numbers are $ 5 $ and $ 2 $.
\[
2x^2 + 7x + 5 = 2x^2 + 5x + 2x + 5 = x(2x + 5) + 1(2x + 5) = (x + 1)(2x + 5)
\]
Answer: $ (x + 1)(2x + 5) $

#### 5) $ 2x^2 + 9x + 7 $
We look for two numbers that multiply to $ 2 \cdot 7 = 14 $ and add to $ 9 $. These numbers are $ 7 $ and $ 2 $.
\[
2x^2 + 9x + 7 = 2x^2 + 7x + 2x + 7 = x(2x + 7) + 1(2x + 7) = (x + 1)(2x + 7)
\]
Answer: $ (x + 1)(2x + 7) $

#### 6) $ 2x^2 + 5x + 3 $
We look for two numbers that multiply to $ 2 \cdot 3 = 6 $ and add to $ 5 $. These numbers are $ 3 $ and $ 2 $.
\[
2x^2 + 5x + 3 = 2x^2 + 3x + 2x + 3 = x(2x + 3) + 1(2x + 3) = (x + 1)(2x + 3)
\]
Answer: $ (x + 1)(2x + 3) $

#### 7) $ 2x^2 + 8x + 6 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 8x + 6 = 2(x^2 + 4x + 3)
\]
Now, factor $ x^2 + 4x + 3 $. We look for two numbers that multiply to $ 3 $ and add to $ 4 $. These numbers are $ 3 $ and $ 1 $.
\[
x^2 + 4x + 3 = (x + 3)(x + 1)
\]
So,
\[
2x^2 + 8x + 6 = 2(x + 3)(x + 1)
\]
Answer: $ 2(x + 3)(x + 1) $

#### 8) $ 2x^2 + 9x + 10 $
We look for two numbers that multiply to $ 2 \cdot 10 = 20 $ and add to $ 9 $. These numbers are $ 5 $ and $ 4 $.
\[
2x^2 + 9x + 10 = 2x^2 + 5x + 4x + 10 = x(2x + 5) + 2(2x + 5) = (x + 2)(2x + 5)
\]
Answer: $ (x + 2)(2x + 5) $

#### 9) $ 2x^2 + 16x + 14 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 16x + 14 = 2(x^2 + 8x + 7)
\]
Now, factor $ x^2 + 8x + 7 $. We look for two numbers that multiply to $ 7 $ and add to $ 8 $. These numbers are $ 7 $ and $ 1 $.
\[
x^2 + 8x + 7 = (x + 7)(x + 1)
\]
So,
\[
2x^2 + 16x + 14 = 2(x + 7)(x + 1)
\]
Answer: $ 2(x + 7)(x + 1) $

#### 10) $ 2x^2 + 16x + 24 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 16x + 24 = 2(x^2 + 8x + 12)
\]
Now, factor $ x^2 + 8x + 12 $. We look for two numbers that multiply to $ 12 $ and add to $ 8 $. These numbers are $ 6 $ and $ 2 $.
\[
x^2 + 8x + 12 = (x + 6)(x + 2)
\]
So,
\[
2x^2 + 16x + 24 = 2(x + 6)(x + 2)
\]
Answer: $ 2(x + 6)(x + 2) $

#### 11) $ 2x^2 + 12x + 18 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 12x + 18 = 2(x^2 + 6x + 9)
\]
Now, factor $ x^2 + 6x + 9 $. This is a perfect square trinomial:
\[
x^2 + 6x + 9 = (x + 3)^2
\]
So,
\[
2x^2 + 12x + 18 = 2(x + 3)^2
\]
Answer: $ 2(x + 3)^2 $

#### 12) $ 2x^2 + 14x + 20 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 14x + 20 = 2(x^2 + 7x + 10)
\]
Now, factor $ x^2 + 7x + 10 $. We look for two numbers that multiply to $ 10 $ and add to $ 7 $. These numbers are $ 5 $ and $ 2 $.
\[
x^2 + 7x + 10 = (x + 5)(x + 2)
\]
So,
\[
2x^2 + 14x + 20 = 2(x + 5)(x + 2)
\]
Answer: $ 2(x + 5)(x + 2) $

#### 13) $ 2x^2 + 22x + 36 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 22x + 36 = 2(x^2 + 11x + 18)
\]
Now, factor $ x^2 + 11x + 18 $. We look for two numbers that multiply to $ 18 $ and add to $ 11 $. These numbers are $ 9 $ and $ 2 $.
\[
x^2 + 11x + 18 = (x + 9)(x + 2)
\]
So,
\[
2x^2 + 22x + 36 = 2(x + 9)(x + 2)
\]
Answer: $ 2(x + 9)(x + 2) $

#### 14) $ 2x^2 + 28x + 48 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 28x + 48 = 2(x^2 + 14x + 24)
\]
Now, factor $ x^2 + 14x + 24 $. We look for two numbers that multiply to $ 24 $ and add to $ 14 $. These numbers are $ 12 $ and $ 2 $.
\[
x^2 + 14x + 24 = (x + 12)(x + 2)
\]
So,
\[
2x^2 + 28x + 48 = 2(x + 12)(x + 2)
\]
Answer: $ 2(x + 12)(x + 2) $

#### 15) $ 2x^2 + 26x + 72 $
First, factor out the GCF, which is $ 2 $:
\[
2x^2 + 26x + 72 = 2(x^2 + 13x + 36)
\]
Now, factor $ x^2 + 13x + 36 $. We look for two numbers that multiply to $ 36 $ and add to $ 13 $. These numbers are $ 9 $ and $ 4 $.
\[
x^2 + 13x + 36 = (x + 9)(x + 4)
\]
So,
\[
2x^2 + 26x + 72 = 2(x + 9)(x + 4)
\]
Answer: $ 2(x + 9)(x + 4) $

---

Final Answer:

\[
\boxed{
\begin{array}{lll}
\text{Section A:} & \text{Section B:} & \text{Section C:} \\
1) (2x - 1)(x + 1) & 1) (x + 10)(x - 3) & 1) (2x + 1)(x + 1) \\
2) (x - 1)(2x + 3) & 2) (x + 5)(x + 4) & 2) (2x + 1)(x + 2) \\
3) (2x - 1)(x + 5) & 3) (x + 9)(x - 1) & 3) (2x + 1)(x + 3) \\
4) (2x + 1)(x - 2) & 4) (x - 10)(x - 8) & 4) (x + 1)(2x + 5) \\
5) (2x + 3)(x - 8) & 5) (x - 7)(x - 4) & 5) (x + 1)(2x + 7) \\
6) (3x + 1)(x - 5) & 6) (x + 12)(x - 6) & 6) (x + 1)(2x + 3) \\
7) (x + 1)(3x - 11) & 7) (x - 11)(x + 2) & 7) 2(x + 3)(x + 1) \\
8) 2(x - 6)(x - 1) & 8) (x - 4)(x + 3) & 8) (x + 2)(2x + 5) \\
9) 3(x - 4)(x - 3) & 9) (x + 12)(x - 9) & 9) 2(x + 7)(x + 1) \\
10) (5x - 1)(x - 8) & 10) (x - 8)(x - 9) & 10) 2(x + 6)(x + 2) \\
11) (3x + 7)(x - 3) & 11) (x - 7)(x + 6) & 11) 2(x + 3)^2 \\
12) 2(x + 3)(x - 2) & 12) (x - 7)(x - 8) & 12) 2(x + 5)(x + 2) \\
13) (2x - 5)(x - 3) & & 13) 2(x + 9)(x + 2) \\
14) (3x + 2)(x - 12) & & 14) 2(x + 12)(x + 2) \\
15) (5x - 2)(x - 5) & & 15) 2(x + 9)(x + 4) \\
\end{array}
}
\]

Parent Tip: Review the logic above to help your child master the concept of math 9th grade worksheet.

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