Problem 114 Simplify. $$-2\left(y^{2}-2 y+... [FREE SOLUTION]

Chapter 4: Problem 114

Simplify. $$-2\left(y^{2}-2 y+4\right)$$

Short Answer

Expert verified

The simplified form of the expression is $-2y^2 + 4y - 8$.

Step by step solution

Distribute the -2

In this step, distribute the -2 into each term inside the parenthesis. This means you multiply -2 by each of the terms: $y^2$, $-2y$, and 4, respectively.

Perform Multiplication

Now, multiply -2 by each of the terms. This equals: $-2 * y^2 = -2y^2$, $-2 * -2y = 4y$, and $-2 * 4 = -8$. Now, combine these results to form the new expression.

Write Down the Final Answer

Now, write down the new, simplified expression: -2y^2 + 4y - 8.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions by removing parentheses. Imagine you have an expression like $-2(y^2 - 2y + 4)$.

Here, the distributive property allows you to "distribute" the $-2$ across each term inside the parentheses. It's like sharing equally among each term. Therefore, you multiply $-2$ by each of the terms inside: $y^2$, $-2y$, and $4$.

Multiply $-2$ and $y^2$ to get $-2y^2$.
Multiply $-2$ and $-2y$ giving you $4y$ since two negatives make a positive.
Lastly, $-2$ times $4$ gives $-8$.

Using the distributive property correctly breaks down the problem, making it much simpler to handle.

Mastering Negative Multiplication

When dealing with negative numbers, understanding how they interact with others is crucial. The original expression $-2(y^2 - 2y + 4)$ requires us to multiply $-2$ with every term inside the parentheses.

This multiplication embodies the rules of negative numbers:

Negative times Positive: Multiplying $-2$ by $y^2$ gives $-2y^2$. The result remains negative.
Negative times Negative: When $-2$ is multiplied by $-2y$, it produces a positive $4y$. Remember, two negatives always yield a positive.
Negative times Positive: Lastly, $-2$ by $4$ results in $-8$.

Mastering these rules simplifies problems efficiently and avoids mistakes.

Combining Like Terms

Once the distributive property and multiplication are complete, the next step is simplifying by combining like terms.

"Like terms" are those terms in an expression that contain the same variable raised to the same power. In our expression $-2y^2 + 4y - 8$, each part is unique in form:

$-2y^2$ is a term involving $y^2$.
$4y$ contains the variable $y$.
$-8$ is a standalone constant.

No like terms exist to combine here, so the simplified expression remains the same. Recognizing like terms quickly can greatly reduce complexity in larger problems.

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91影视

Simplify. $$-2\left(y^{2}-2 y+4\right)$$

Short Answer

Step by step solution

Distribute the -2

Perform Multiplication

Write Down the Final Answer

Key Concepts

Understanding the Distributive Property

Mastering Negative Multiplication

Combining Like Terms

One App. One Place for Learning.

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