Question - Simplifying Expressions Using the Distributive Property
Solution:
1. Apply the distributive property to 3p(p - q): 3p * p - 3p * q = 3p^2 - 3pq2. Expand the square of the binomial (2p - q)^2: (2p - q)(2p - q) = 4p^2 - 4pq + q^23. Subtract the expanded binomial from the first expression: (3p^2 - 3pq) - (4p^2 - 4pq + q^2)4. Distribute the negative sign to each term in the second expression: 3p^2 - 3pq - 4p^2 + 4pq - q^25. Combine like terms: -p^2 + pq - q^2So the simplified expression is: -p^2 + pq - q^2
Tutor Solution:
The image shows a mathematical expression that needs simplifying. Let's go through the simplifying process step by step.The expression is:\[3p(p - q) - (2p - q)^2\]Step 1: Distribute the first term.The first term is a product of $$3p$$ and $$(p - q)$$. Use the distributive property of multiplication over subtraction to expand this:\[3p(p) - 3p(q) = 3p^2 - 3pq\]Step 2: Expand the squared term.The second term is a square of the binomial $$(2p - q)$$. The square of a binomial $$(a - b)^2$$ is $$a^2 - 2ab + b^2$$. Applying this pattern, we get:\[(2p - q)^2 = (2p)^2 - 2 \cdot (2p) \cdot q + q^2 = 4p^2 - 4pq + q^2\]Step 3: Combine the expanded terms.Substitute our expansions from steps 1 and 2 into the original equation:\[3p^2 - 3pq - (4p^2 - 4pq + q^2)\]Step 4: Distribute the negative sign over the expanded squared term.When subtracting the expanded squared term, you have to apply the negative sign to each term within the parentheses:\[3p^2 - 3pq - 4p^2 + 4pq - q^2\]Step 5: Combine like terms.Now, combine the terms with $$p^2$$, the terms with $$pq$$, and the constant terms separately:\[(3p^2 - 4p^2) + (-3pq + 4pq) - q^2\]Doing the operations gives us:\[-p^2 + pq - q^2\]This is the simplified form of the original expression. The result is:\[-p^2 + pq - q^2\]Remember, in math, combining like terms and using algebraic properties to simplify expressions are crucial techniques. Always look for opportunities to group and combine similar terms and apply the basic algebraic rules to arrive at a simplified result.
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