To convert the expression \(x^2 + 14\) into a perfect square trinomial, we need to determine the appropriate constant to add.
A perfect square trinomial takes the form \((x + a)^2\) which expands to \(x^2 + 2ax + a^2\).
From your given expression \(x^2 + 14\), we can identify that it currently lacks the linear term \(2ax\).
To make this a perfect square trinomial, we first need to establish our goal, which is to express \(x^2 + 14 + k\) in the form of a perfect square. Adding a term \(k\) will create a trinomial which will be of the form \(x^2 + 14 + k = (x + b)^2\) for some \(b\).
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First, rewrite the expression: \[ x^2 + 14 + k \]
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Set the right part equal to the squared binomial: \[ x^2 + 14 + k = (x + b)^2 = x^2 + 2bx + b^2 \]
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Equating the constant terms: To achieve this, we match the constant terms: \[ 14 + k = b^2 \] Hence, the equation we need is: \[ k = b^2 - 14 \]
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Choosing a value of \(b\): To make calculations simple, let's choose \(b = 4\): \[ k = 4^2 - 14 = 16 - 14 = 2 \]
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Adding this back to our original expression: Therefore, we need to add 2 to get: \[ x^2 + 14 + 2 = x^2 + 16 \]
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Expressing as a perfect square: The expression \(x^2 + 16\) can now be expressed as: \[ x^2 + 16 = x^2 + 2(4)x + 4^2 = (x + 4)^2 \]
Thus, the number you need to add to \(x^2 + 14\) to make it a perfect square trinomial is \(2\), and the trinomial can be written as the square of a binomial:
\[ (x + 4)^2 \]
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