Rewrite the expression [tex]x^2+14x + 49[/tex] by applying an appropriate structure.
Answer (2)
Recognize the expression as a quadratic trinomial.
Factor the quadratic expression by finding two numbers that add up to 14 and multiply to 49, which are both 7.
Rewrite the expression as ( x + 7 ) ( x + 7 ) .
Simplify the expression to ( x + 7 ) 2 . The expression x 2 + 14 x + 49 can be rewritten as ( x + 7 ) 2 .
Explanation
Understanding the Expression We are given the expression 2 + 14 x + 49 and asked to rewrite it in an appropriate structure. It seems there's a typo in the question, and the expression should be x 2 + 14 x + 49 . Assuming the expression is x 2 + 14 x + 49 , we can proceed to rewrite it.
Factoring the Quadratic We observe that the expression x 2 + 14 x + 49 is a quadratic expression. We can try to factor it. We are looking for two numbers that add up to 14 and multiply to 49. Since 7 + 7 = 14 and 7 × 7 = 49 , we can rewrite the expression as ( x + 7 ) ( x + 7 ) .
Rewriting the Expression Since ( x + 7 ) ( x + 7 ) is the same as ( x + 7 ) 2 , we can rewrite the expression as ( x + 7 ) 2 .
Examples
Factoring quadratic expressions is useful in many real-world applications. For example, if you are designing a rectangular garden and you know the area is given by the expression x 2 + 14 x + 49 , you can factor it to find the dimensions of the garden. In this case, the garden would be a square with side length x + 7 . This helps in planning the layout and determining the amount of fencing needed.
The expression x 2 + 14 x + 49 can be rewritten as ( x + 7 ) 2 by recognizing it as a perfect square trinomial. This involves factoring the quadratic expression to find two identical binomials. The process confirms that x 2 + 14 x + 49 = ( x + 7 ) ( x + 7 ) = ( x + 7 ) 2 .
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