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Advanced Math / Equivalent expressions Difficulty: Medium

$x squared, plus 6 x, plus 4$

Which of the following is equivalent to the expression above?

(x + 3)² + 5

(x + 3)² – 5

(x – 3)² + 5

(x – 3)² – 5

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Explanation

Choice B is correct. The given quadratic expression is in standard form, and each answer choice is in vertex form. Completing the square converts the expression from standard form to vertex form. The first step is to rewrite the expression as follows: $x squared, plus 6 x, plus 4, equals, x squared, plus 6 x, plus 9, plus 4, minus 9$ . The first three terms of the revised expression can be rewritten as a perfect square as follows: $x squared, plus 6 x, plus 9, plus 4, minus 9, equals, open parenthesis, x plus 3, close parenthesis, squared, plus 4, minus 9$ . Combining the constant terms gives $open parenthesis, x plus 3, close parenthesis, squared, minus 5$ .

Choice A is incorrect. Squaring the binomial and simplifying the expression in choice A gives $x squared, plus 6 x, plus 9, plus 5$ . Combining like terms gives $x squared, minus 6 x, plus 14$ , not $x squared, plus 6 x, plus 4$ . Choice C is incorrect. Squaring the binomial and simplifying the expression in choice C gives $x squared, minus 6 x, plus 9, plus 5$ . Combining like terms gives $x squared, minus 6 x, plus 14$ , not $x squared, plus 6 x, plus 4$ . Choice D is incorrect. Squaring the binomial and simplifying the expression in choice D gives $x squared, minus 6 x, plus 9, minus 5$ . Combining like terms gives $x squared, minus 6 x, plus 4$ , not $x squared, plus 6 x, plus 4$ .