Math 9 Special Products Square of a Binomial online exercise for ... - Free Printable

Let's solve each of the square binomials using the square of a binomial formula:

$$
(a \pm b)^2 = a^2 \pm 2ab + b^2
$$

We'll apply this formula step by step to each expression.

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1. $(x + 5)^2$

Use the formula:
$$
(a + b)^2 = a^2 + 2ab + b^2
$$
Here, $a = x$, $b = 5$

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

✔ Answer: $x^2 + 10x + 25$

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2. $(2x - 5)^2$

Use the formula:
$$
(a - b)^2 = a^2 - 2ab + b^2
$$
Here, $a = 2x$, $b = 5$

$$
(2x - 5)^2 = (2x)^2 - 2(2x)(5) + 5^2 = 4x^2 - 20x + 25
$$

✔ Answer: $4x^2 - 20x + 25$

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3. $(3x - 2)^2$

Again, use $(a - b)^2 = a^2 - 2ab + b^2$
Here, $a = 3x$, $b = 2$

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

✔ Answer: $9x^2 - 12x + 4$

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4. $\left(x^2 - \frac{1}{2}x\right)^2$

This is a bit more complex, but we still use the same formula:
$$
(a - b)^2 = a^2 - 2ab + b^2
$$
Here, $a = x^2$, $b = \frac{1}{2}x$

Now compute each term:

- $a^2 = (x^2)^2 = x^4$
- $2ab = 2(x^2)\left(\frac{1}{2}x\right) = 2 \cdot \frac{1}{2} \cdot x^2 \cdot x = 1 \cdot x^3 = x^3$
- $b^2 = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$

So:
$$
\left(x^2 - \frac{1}{2}x\right)^2 = x^4 - x^3 + \frac{1}{4}x^2
$$

✔ Answer: $x^4 - x^3 + \frac{1}{4}x^2$

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✔ Final Answers:

1. $(x + 5)^2 = x^2 + 10x + 25$
2. $(2x - 5)^2 = 4x^2 - 20x + 25$
3. $(3x - 2)^2 = 9x^2 - 12x + 4$
4. $\left(x^2 - \frac{1}{2}x\right)^2 = x^4 - x^3 + \frac{1}{4}x^2$

Let me know if you'd like to see the steps visually or with diagrams!