Perfect squares expansion worksheet (with solutions) | Teaching ... - Free Printable

Let's solve each of the perfect square expansions step by step using the perfect square formula:

🔷 Perfect Square Formula:

For any expression $(a + b)^2$, the expansion is:
$$
(a + b)^2 = a^2 + 2ab + b^2
$$

Similarly,
$$
(a - b)^2 = a^2 - 2ab + b^2
$$

We’ll apply this to each problem.

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15) $(6 + 2x)^2$

Let $a = 6$, $b = 2x$

$$
(6 + 2x)^2 = 6^2 + 2(6)(2x) + (2x)^2 = 36 + 24x + 4x^2
$$

✔ Answer: $4x^2 + 24x + 36$

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16) $(7 - 3x)^2$

Let $a = 7$, $b = 3x$

$$
(7 - 3x)^2 = 7^2 - 2(7)(3x) + (3x)^2 = 49 - 42x + 9x^2
$$

✔ Answer: $9x^2 - 42x + 49$

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17) $(10 + 4y)^2$

Let $a = 10$, $b = 4y$

$$
(10 + 4y)^2 = 10^2 + 2(10)(4y) + (4y)^2 = 100 + 80y + 16y^2
$$

✔ Answer: $16y^2 + 80y + 100$

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18) $(x + y)^2$

This is the basic identity.

$$
(x + y)^2 = x^2 + 2xy + y^2
$$

✔ Answer: $x^2 + 2xy + y^2$

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19) $(x - 2y)^2$

Let $a = x$, $b = 2y$

$$
(x - 2y)^2 = x^2 - 2(x)(2y) + (2y)^2 = x^2 - 4xy + 4y^2
$$

✔ Answer: $x^2 - 4xy + 4y^2$

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20) $(2x + 3y)^2$

Let $a = 2x$, $b = 3y$

$$
(2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 = 4x^2 + 12xy + 9y^2
$$

✔ Answer: $4x^2 + 12xy + 9y^2$

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21) $(x^2 + 5)^2$

Let $a = x^2$, $b = 5$

$$
(x^2 + 5)^2 = (x^2)^2 + 2(x^2)(5) + 5^2 = x^4 + 10x^2 + 25
$$

✔ Answer: $x^4 + 10x^2 + 25$

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22) $(4 - y^2)^2$

Let $a = 4$, $b = y^2$

$$
(4 - y^2)^2 = 4^2 - 2(4)(y^2) + (y^2)^2 = 16 - 8y^2 + y^4
$$

✔ Answer: $y^4 - 8y^2 + 16$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 15) $(6 + 2x)^2$ | $4x^2 + 24x + 36$ |
| 16) $(7 - 3x)^2$ | $9x^2 - 42x + 49$ |
| 17) $(10 + 4y)^2$ | $16y^2 + 80y + 100$ |
| 18) $(x + y)^2$ | $x^2 + 2xy + y^2$ |
| 19) $(x - 2y)^2$ | $x^2 - 4xy + 4y^2$ |
| 20) $(2x + 3y)^2$ | $4x^2 + 12xy + 9y^2$ |
| 21) $(x^2 + 5)^2$ | $x^4 + 10x^2 + 25$ |
| 22) $(4 - y^2)^2$ | $y^4 - 8y^2 + 16$ |

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