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Advanced Math / Equivalent expressions Difficulty: Hard

$0.36 x^{2} + 0.63 x + 1.17$

The given expression can be rewritten as $a (4 x^{2} + 7 x + 13)$ , where $a$ is a constant. What is the value of $a$ ?

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Explanation

The correct answer is $.09$ . It's given that the expression $0.36 x^{2} + 0.63 x + 1.17$ can be rewritten as $a (4 x^{2} + 7 x + 13)$ . Applying the distributive property to the expression $a (4 x^{2} + 7 x + 13)$ yields $4 a x^{2} + 7 a x + 13 a$ . Therefore, $0.36 x^{2} + 0.63 x + 1.17$ can be rewritten as $4 a x^{2} + 7 a x + 13 a$ . It follows that in the expressions $0.36 x^{2} + 0.63 x + 1.17$ and $4 a x^{2} + 7 a x + 13 a$ , the coefficients of $x^{2}$ are equivalent, the coefficients of $x$ are equivalent, and the constant terms are equivalent. Therefore, $0.36 equals 4 a$ , $0.63 equals 7 a$ , and $1.17 equals 13 a$ . Solving any of these equations for $a$ yields the value of $a$ . Dividing both sides of the equation $0.36 equals 4 a$ by $4$ yields $0.09 equals a$ . Therefore, the value of $a$ is $0.09$ . Note that .09 and 9/100 are examples of ways to enter a correct answer.