To rewrite the function \( f(x) = 2x^2 + 13x + 20 \) by completing the square, follow these steps:
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Factor out the coefficient of \( x^2 \) (which is 2) from the first two terms: \[ f(x) = 2(x^2 + \frac{13}{2}x) + 20 \]
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Complete the square for the expression in parentheses \( x^2 + \frac{13}{2}x \). To do this, take half of the coefficient of \( x \) (which is \( \frac{13}{2} \)), square it, and add and subtract it inside the parentheses.
Half of \( \frac{13}{2} \) is \( \frac{13}{4} \). The square of \( \frac{13}{4} \) is \( \left(\frac{13}{4}\right)^2 = \frac{169}{16} \).
Then we can rewrite the expression: \[ x^2 + \frac{13}{2}x = \left(x + \frac{13}{4}\right)^2 - \frac{169}{16} \]
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Substitute this back into the function: \[ f(x) = 2\left(\left(x + \frac{13}{4}\right)^2 - \frac{169}{16}\right) + 20 \]
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Distribute the 2: \[ f(x) = 2\left(x + \frac{13}{4}\right)^2 - 2\cdot\frac{169}{16} + 20 \] \[ = 2\left(x + \frac{13}{4}\right)^2 - \frac{338}{16} + 20 \]
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Convert \( 20 \) into sixteenths to combine the constants: \[ 20 = \frac{320}{16} \]
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Combine the constants: \[ -\frac{338}{16} + \frac{320}{16} = -\frac{18}{16} = -\frac{9}{8} \]
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Finally, we have the completed square form: \[ f(x) = 2\left(x + \frac{13}{4}\right)^2 - \frac{9}{8} \]
Thus, the completed square form is: \[ f(x) = 2\left( x + \frac{13}{4} \right)^2 - \frac{9}{8} \]
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