Which is equivalent to (4xy - 3z)², and what type of special product is it?
Understand the Problem
The question is asking which expression is equivalent to (4xy - 3z)² and what type of special product it represents. It involves understanding formulas related to algebraic identities.
Answer
The equivalent expression is $16x^2y^2 - 24xyz + 9z^2$, a perfect square trinomial.
Answer for screen readers
The expression equivalent to $(4xy - 3z)^2$ is $16x^2y^2 - 24xyz + 9z^2$, and it is a perfect square trinomial.
Steps to Solve
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Identify the expression We are given the expression $(4xy - 3z)^2$. We need to find an equivalent expression and identify what type of special product it represents.
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Apply the square of a binomial formula Recall the formula for the square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$ In this case, let $a = 4xy$ and $b = 3z$.
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Calculate $a^2$ and $b^2$ Calculate $a^2$: $$a^2 = (4xy)^2 = 16x^2y^2.$$
Calculate $b^2$: $$b^2 = (3z)^2 = 9z^2.$$
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Calculate the middle term $-2ab$ Now calculate the middle term: $$-2ab = -2(4xy)(3z) = -24xyz.$$
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Combine the results Now, combine the results into the formula for the square of a binomial: $$(4xy - 3z)^2 = 16x^2y^2 - 24xyz + 9z^2.$$
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Determine the type of special product The result $16x^2y^2 - 24xyz + 9z^2$ represents a perfect square trinomial.
The expression equivalent to $(4xy - 3z)^2$ is $16x^2y^2 - 24xyz + 9z^2$, and it is a perfect square trinomial.
More Information
A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form is $a^2 - 2ab + b^2$, which corresponds to $(a - b)^2$.
Tips
- Confusing the formula for the difference of squares and the square of a binomial. Remember that $(a - b)^2$ includes a middle term that is negative.
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