Problem 100 Simplify. $$ 2\left(x^{2}-3\... [FREE SOLUTION]

Chapter 1: Problem 100

Simplify. $$ 2\left(x^{2}-3\right)-\left(6 x^{2}-2\right)+2\left(-7 x^{2}-4\right) $$

Short Answer

Expert verified

-18x^{2} - 12

Step by step solution

- Distribute the constants

Multiply each term inside the parentheses by the constants outside. 2(x^{2} - 3) = 2x^{2} - 6 -1(6x^{2} - 2) = -6x^{2} + 2 2(-7x^{2} - 4) = -14x^{2} - 8

- Combine the like terms

Add all the terms with x^{2} and all the constant terms separately. 2x^{2} - 6 - 6x^{2} + 2 - 14x^{2} - 8 = (2x^{2} - 6x^{2} - 14x^{2}) + (-6 + 2 - 8)

- Simplify the expression

Simplify the expression by performing the additions and subtractions: 2x^{2} - 6x^{2} - 14x^{2} = -18x^{2} -6 + 2 - 8 = -12 Therefore, the simplified expression is -18x^{2} - 12.

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Key Concepts

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

headline of the respective core concept

Distribution is the first step we take when simplifying expressions like the one provided. In this case, distribution means multiplying each term within the parentheses by the constant outside the parentheses.

Let's break it down. In the expression, we have: $2(x^{2} - 3)$, $-1(6x^{2} - 2)$, and $2(-7x^{2} - 4)$.

For $2(x^{2} - 3)$, we multiply 2 by both $x^{2}$ and -3, giving us $2x^{2} - 6$.
For $-1(6x^{2} - 2)$, -1 is multiplied by 6$x^{2}$ and -2, resulting in $-6x^{2} + 2$.
For $2(-7x^{2} - 4)$, 2 is multiplied by both -7$x^{2}$ and -4, leading to $-14x^{2} - 8$.

This step ensures that each term inside the parentheses is accounted for separately, before moving on to the next step.

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Once distribution is complete, the next crucial step is combining like terms. Like terms are terms that have the same variable raised to the same power. Here, we group all the terms with $x^{2}$ and all the constants.

After distribution, we have $2x^{2} - 6 - 6x^{2} + 2 - 14x^{2} - 8$.

Group the $x^{2}$ terms: $2x^{2} - 6x^{2} - 14x^{2}$
Group the constants: $-6 + 2 - 8$

Chaining them together helps in clearly identifying and combining similar elements.

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Finally, we perform simplification in algebra, which means combining the grouped like terms. Simplification reduces the expression to its simplest form.

We simplify the $x^{2}$ terms first: $2x^{2} - 6x^{2} - 14x^{2} = -18x^{2}$.

Then we simplify the constants: $-6 + 2 - 8 = -12$.

Combining these simplified parts, we get the final simplified form: $-18x^{2} - 12$.

This simplified expression is much easier to work with and represents the same value as the original expression.

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Simplify. $$ 2\left(x^{2}-3\right)-\left(6 x^{2}-2\right)+2\left(-7 x^{2}-4\right) $$

Short Answer

Step by step solution

- Distribute the constants

- Combine the like terms

- Simplify the expression

Key Concepts

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

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