7th Grade Math Worksheets PDF | Printable Worksheets - Free Printable

Problem: Multiplying Binomials

The task is to expand the given binomials and simplify the resulting expressions. We will solve a few examples from each section to illustrate the process.

---

#### Section A: Expanding Binomials with Variables

The general method for multiplying two binomials is the FOIL method:
- F: First terms
- O: Outer terms
- I: Inner terms
- L: Last terms

After applying FOIL, combine like terms to simplify the expression.

##### Example 1: Expand \((x + 2)(x + 7)\)
\[
(x + 2)(x + 7) = x \cdot x + x \cdot 7 + 2 \cdot x + 2 \cdot 7
\]
\[
= x^2 + 7x + 2x + 14
\]
\[
= x^2 + 9x + 14
\]

##### Example 2: Expand \((x + 9)(x + 5)\)
\[
(x + 9)(x + 5) = x \cdot x + x \cdot 5 + 9 \cdot x + 9 \cdot 5
\]
\[
= x^2 + 5x + 9x + 45
\]
\[
= x^2 + 14x + 45
\]

##### Example 3: Expand \((x - 8)(x + 1)\)
\[
(x - 8)(x + 1) = x \cdot x + x \cdot 1 + (-8) \cdot x + (-8) \cdot 1
\]
\[
= x^2 + x - 8x - 8
\]
\[
= x^2 - 7x - 8
\]

##### Example 4: Expand \((x - 3)(x - 4)\)
\[
(x - 3)(x - 4) = x \cdot x + x \cdot (-4) + (-3) \cdot x + (-3) \cdot (-4)
\]
\[
= x^2 - 4x - 3x + 12
\]
\[
= x^2 - 7x + 12
\]

##### Example 5: Expand \((8x + 10)(x + 3)\)
\[
(8x + 10)(x + 3) = 8x \cdot x + 8x \cdot 3 + 10 \cdot x + 10 \cdot 3
\]
\[
= 8x^2 + 24x + 10x + 30
\]
\[
= 8x^2 + 34x + 30
\]

##### Example 6: Expand \((2x + 11)(9x + 2)\)
\[
(2x + 11)(9x + 2) = 2x \cdot 9x + 2x \cdot 2 + 11 \cdot 9x + 11 \cdot 2
\]
\[
= 18x^2 + 4x + 99x + 22
\]
\[
= 18x^2 + 103x + 22
\]

##### Example 7: Expand \((5x - 4)(x + 3)\)
\[
(5x - 4)(x + 3) = 5x \cdot x + 5x \cdot 3 + (-4) \cdot x + (-4) \cdot 3
\]
\[
= 5x^2 + 15x - 4x - 12
\]
\[
= 5x^2 + 11x - 12
\]

##### Example 8: Expand \((3x + 1)(x - 2)\)
\[
(3x + 1)(x - 2) = 3x \cdot x + 3x \cdot (-2) + 1 \cdot x + 1 \cdot (-2)
\]
\[
= 3x^2 - 6x + x - 2
\]
\[
= 3x^2 - 5x - 2
\]

##### Example 9: Expand \((7x - 4)(4x - 5)\)
\[
(7x - 4)(4x - 5) = 7x \cdot 4x + 7x \cdot (-5) + (-4) \cdot 4x + (-4) \cdot (-5)
\]
\[
= 28x^2 - 35x - 16x + 20
\]
\[
= 28x^2 - 51x + 20
\]

##### Example 10: Expand \((6x - 1)(4x + 3)\)
\[
(6x - 1)(4x + 3) = 6x \cdot 4x + 6x \cdot 3 + (-1) \cdot 4x + (-1) \cdot 3
\]
\[
= 24x^2 + 18x - 4x - 3
\]
\[
= 24x^2 + 14x - 3
\]

##### Example 11: Expand \((8x - 3)(4x + 1)\)
\[
(8x - 3)(4x + 1) = 8x \cdot 4x + 8x \cdot 1 + (-3) \cdot 4x + (-3) \cdot 1
\]
\[
= 32x^2 + 8x - 12x - 3
\]
\[
= 32x^2 - 4x - 3
\]

##### Example 12: Expand \((2x - 6)(3x - 5)\)
\[
(2x - 6)(3x - 5) = 2x \cdot 3x + 2x \cdot (-5) + (-6) \cdot 3x + (-6) \cdot (-5)
\]
\[
= 6x^2 - 10x - 18x + 30
\]
\[
= 6x^2 - 28x + 30
\]

##### Example 13: Expand \((2x - 6)(3x - 5)\)
This is the same as Example 12, so the result is:
\[
6x^2 - 28x + 30
\]

##### Example 14: Expand \((6 - x)(x - 7)\)
\[
(6 - x)(x - 7) = 6 \cdot x + 6 \cdot (-7) + (-x) \cdot x + (-x) \cdot (-7)
\]
\[
= 6x - 42 - x^2 + 7x
\]
\[
= -x^2 + 13x - 42
\]

##### Example 15: Expand \((3 - 2x)(2 - x)\)
\[
(3 - 2x)(2 - x) = 3 \cdot 2 + 3 \cdot (-x) + (-2x) \cdot 2 + (-2x) \cdot (-x)
\]
\[
= 6 - 3x - 4x + 2x^2
\]
\[
= 2x^2 - 7x + 6
\]

---

#### Section B: Expanding Binomials with Multiple Variables

We use the same FOIL method, but now we have multiple variables.

##### Example 1: Expand \((a + b)(a + b)\)
\[
(a + b)(a + b) = a \cdot a + a \cdot b + b \cdot a + b \cdot b
\]
\[
= a^2 + ab + ab + b^2
\]
\[
= a^2 + 2ab + b^2
\]

##### Example 2: Expand \((3a + b)(2a + b)\)
\[
(3a + b)(2a + b) = 3a \cdot 2a + 3a \cdot b + b \cdot 2a + b \cdot b
\]
\[
= 6a^2 + 3ab + 2ab + b^2
\]
\[
= 6a^2 + 5ab + b^2
\]

##### Example 3: Expand \((5a + 2b)(a + b)\)
\[
(5a + 2b)(a + b) = 5a \cdot a + 5a \cdot b + 2b \cdot a + 2b \cdot b
\]
\[
= 5a^2 + 5ab + 2ab + 2b^2
\]
\[
= 5a^2 + 7ab + 2b^2
\]

##### Example 4: Expand \((4a + b)(5a + 2b)\)
\[
(4a + b)(5a + 2b) = 4a \cdot 5a + 4a \cdot 2b + b \cdot 5a + b \cdot 2b
\]
\[
= 20a^2 + 8ab + 5ab + 2b^2
\]
\[
= 20a^2 + 13ab + 2b^2
\]

##### Example 5: Expand \((6a + 3b)(2a - b)\)
\[
(6a + 3b)(2a - b) = 6a \cdot 2a + 6a \cdot (-b) + 3b \cdot 2a + 3b \cdot (-b)
\]
\[
= 12a^2 - 6ab + 6ab - 3b^2
\]
\[
= 12a^2 - 3b^2
\]

##### Example 6: Expand \((7a - 5b)(a + 4b)\)
\[
(7a - 5b)(a + 4b) = 7a \cdot a + 7a \cdot 4b + (-5b) \cdot a + (-5b) \cdot 4b
\]
\[
= 7a^2 + 28ab - 5ab - 20b^2
\]
\[
= 7a^2 + 23ab - 20b^2
\]

##### Example 7: Expand \((a - 3b)(11a - b)\)
\[
(a - 3b)(11a - b) = a \cdot 11a + a \cdot (-b) + (-3b) \cdot 11a + (-3b) \cdot (-b)
\]
\[
= 11a^2 - ab - 33ab + 3b^2
\]
\[
= 11a^2 - 34ab + 3b^2
\]

##### Example 8: Expand \((4a + 5b)(6a - 9b)\)
\[
(4a + 5b)(6a - 9b) = 4a \cdot 6a + 4a \cdot (-9b) + 5b \cdot 6a + 5b \cdot (-9b)
\]
\[
= 24a^2 - 36ab + 30ab - 45b^2
\]
\[
= 24a^2 - 6ab - 45b^2
\]

##### Example 9: Expand \((x + 6)^2\)
\[
(x + 6)^2 = (x + 6)(x + 6)
\]
\[
= x \cdot x + x \cdot 6 + 6 \cdot x + 6 \cdot 6
\]
\[
= x^2 + 6x + 6x + 36
\]
\[
= x^2 + 12x + 36
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
&\text{Section A:} \\
&1. x^2 + 9x + 14 \\
&2. x^2 + 14x + 45 \\
&3. x^2 - 7x - 8 \\
&4. x^2 - 7x + 12 \\
&5. 8x^2 + 34x + 30 \\
&6. 18x^2 + 103x + 22 \\
&7. 5x^2 + 11x - 12 \\
&8. 3x^2 - 5x - 2 \\
&9. 28x^2 - 51x + 20 \\
&10. 24x^2 + 14x - 3 \\
&11. 32x^2 - 4x - 3 \\
&12. 6x^2 - 28x + 30 \\
&13. 6x^2 - 28x + 30 \\
&14. -x^2 + 13x - 42 \\
&15. 2x^2 - 7x + 6 \\
\\
&\text{Section B:} \\
&1. a^2 + 2ab + b^2 \\
&2. 6a^2 + 5ab + b^2 \\
&3. 5a^2 + 7ab + 2b^2 \\
&4. 20a^2 + 13ab + 2b^2 \\
&5. 12a^2 - 3b^2 \\
&6. 7a^2 + 23ab - 20b^2 \\
&7. 11a^2 - 34ab + 3b^2 \\
&8. 24a^2 - 6ab - 45b^2 \\
&9. x^2 + 12x + 36 \\
\end{aligned}
}
\]