Simplify the algebraic expression: [tex]x(x+3)+x(2 x-4)+6[/tex]
A) [tex]2 x^4-x^2+6[/tex]
B) [tex]3 x^2-x+6[/tex]
C) [tex]3 x^2+x+6[/tex]
D) [tex]3 x^2+5[/tex]
Answer (2)
Expand the first term: x ( x + 3 ) = x 2 + 3 x .
Expand the second term: x ( 2 x − 4 ) = 2 x 2 − 4 x .
Combine all terms: x 2 + 3 x + 2 x 2 − 4 x + 6 .
Simplify the expression: 3 x 2 − x + 6 . The answer is 3 x 2 − x + 6 .
Explanation
Understanding the Problem We are given the expression x ( x + 3 ) + x ( 2 x − 4 ) + 6 and asked to simplify it. We will expand the terms, combine like terms, and then compare our result to the answer choices.
Expanding the First Term First, we expand the first term: x ( x + 3 ) = x 2 + 3 x .
Expanding the Second Term Next, we expand the second term: x ( 2 x − 4 ) = 2 x 2 − 4 x .
Combining the Terms Now, we combine the expanded terms and the constant: x 2 + 3 x + 2 x 2 − 4 x + 6 .
Simplifying the Expression We combine like terms: ( x 2 + 2 x 2 ) + ( 3 x − 4 x ) + 6 = 3 x 2 − x + 6 .
Comparing with Options Finally, we compare our simplified expression 3 x 2 − x + 6 with the given options. We see that it matches option B.
Examples
Simplifying algebraic expressions is a fundamental skill in algebra and is used in various real-life situations. For example, if you are calculating the area of a garden with variable dimensions, you might end up with an expression like the one we simplified. Suppose the length of the garden is x + 3 and another part is 2 x − 4 , and you need to add a constant area of 6 square units for a patio. The total area can be represented by x ( x + 3 ) + x ( 2 x − 4 ) + 6 . Simplifying this expression helps you easily calculate the total area for different values of x .
The expression x ( x + 3 ) + x ( 2 x − 4 ) + 6 simplifies to 3 x 2 − x + 6 , which matches option B. Therefore, the correct choice is option B.
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