Questions: Use the product of conjugates to multiply (6y - 1/2 q^2)(6y + 1/2 q^2).

Transcript text: Use the product of conjugates to multiply $\left(6 y-\frac{1}{2} q^{2}\right)\left(6 y+\frac{1}{2} q^{2}\right)$.

Solution

Solution Steps

To solve this problem, we can use the formula for the product of conjugates: $(a - b)(a + b) = a^2 - b^2$. Here, $a = 6y$ and $b = \frac{1}{2}q^2$. We will apply this formula to find the product.

Step 1: Identify the Expression

We are given the expression $(6y - \frac{1}{2}q^2)(6y + \frac{1}{2}q^2)$. This is a product of conjugates.

Step 2: Apply the Product of Conjugates Formula

The product of conjugates formula is $(a - b)(a + b) = a^2 - b^2$. Here, $a = 6y$ and $b = \frac{1}{2}q^2$.

Step 3: Calculate $a^2$ and $b^2$

Calculate $a^2$: \[ a^2 = (6y)^2 = 36y^2 \]
Calculate $b^2$: \[ b^2 = \left(\frac{1}{2}q^2\right)^2 = \frac{1}{4}q^4 \]

Step 4: Substitute into the Formula

Substitute $a^2$ and $b^2$ into the formula: \[ a^2 - b^2 = 36y^2 - \frac{1}{4}q^4 \]

Final Answer

The product of the conjugates is: \[ \boxed{36y^2 - \frac{1}{4}q^4} \]

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