Which expression is equivalent to
?
The Correct Answer and Explanation is:
The problem is asking:
Which expression is equivalent to−14−35d?-14 – 35d?−14−35d?
Let’s analyze each answer choice and see which one simplifies to −14−35d-14 – 35d−14−35d.
Step 1: Factor −14−35d-14 – 35d−14−35d
We look for a common factor of both terms.
−14=−7⋅2-14 = -7 \cdot 2−14=−7⋅2
−35d=−7⋅5d-35d = -7 \cdot 5d−35d=−7⋅5d
So we factor out -7:−14−35d=−7(2+5d)-14 – 35d = -7(2 + 5d)−14−35d=−7(2+5d)
Step 2: Match with the options
Let’s examine each choice:
- -7(2 + 5d) ✅
This matches our factorization exactly. - 7(2 − 5d)
Distributing gives 14−35d14 – 35d14−35d, which is not the same. - 7(−2 + 5d)
Distributing gives −14+35d-14 + 35d−14+35d, which is also not correct. - -7(2 − 5d)
Distributing gives −14+35d-14 + 35d−14+35d, which is also not correct.
✅ Correct Answer:
-7(2 + 5d)
Explanation:
To find which expression is equivalent to −14−35d-14 – 35d−14−35d, we use factoring. Factoring is a technique that rewrites a mathematical expression as a product of its greatest common factor (GCF) and a remaining expression inside parentheses.
In this case, we look at the terms −14-14−14 and −35d-35d−35d. Both numbers have a common factor of −7-7−7:
- −14=−7×2-14 = -7 \times 2−14=−7×2
- −35d=−7×5d-35d = -7 \times 5d−35d=−7×5d
So we can factor −7-7−7 out of both terms:−14−35d=−7(2+5d)-14 – 35d = -7(2 + 5d)−14−35d=−7(2+5d)
This tells us that the expression is equivalent to -7(2 + 5d).
Let’s check this by distributing:−7(2+5d)=−7⋅2+(−7⋅5d)=−14−35d-7(2 + 5d) = -7 \cdot 2 + (-7 \cdot 5d) = -14 – 35d−7(2+5d)=−7⋅2+(−7⋅5d)=−14−35d
This confirms the factorization is correct.
The other options do not match because when you distribute them, you get different signs or values. For example, 7(2−5d)7(2 – 5d)7(2−5d) becomes 14−35d14 – 35d14−35d, which is not the same as −14−35d-14 – 35d−14−35d.
In conclusion, factoring helps simplify and understand algebraic expressions by identifying common factors. The correct choice is the one that, when expanded, gives the original expression. Thus, -7(2 + 5d) is the correct and equivalent expression.
