To rewrite the quadratic function \( y = x^2 + 3x + 4 \) in vertex form (which is \( y = a(x - h)^2 + k \)), we will complete the square.
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Start with the original function: \[ y = x^2 + 3x + 4 \]
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Focus on the \( x^2 + 3x \) part. To complete the square, take the coefficient of \( x \) (which is 3), divide it by 2 (gives \( \frac{3}{2} \)), and then square it (gives \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \)).
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Rewrite the equation by adding and subtracting \( \frac{9}{4} \): \[ y = x^2 + 3x + \frac{9}{4} - \frac{9}{4} + 4 \]
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This can be reorganized as: \[ y = \left(x^2 + 3x + \frac{9}{4}\right) + 4 - \frac{9}{4} \]
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The expression in the parentheses is a perfect square: \[ y = \left(x + \frac{3}{2}\right)^2 + 4 - \frac{9}{4} \]
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Now combine \( 4 \) and \( -\frac{9}{4} \). Convert 4 to fractions with the same denominator: \[ 4 = \frac{16}{4} \]
Thus, \[ 4 - \frac{9}{4} = \frac{16}{4} - \frac{9}{4} = \frac{7}{4} \]
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Finally, express the function in vertex form: \[ y = \left(x + \frac{3}{2}\right)^2 + \frac{7}{4} \]
So, the vertex form of the quadratic function \( y = x^2 + 3x + 4 \) is: \[ y = \left(x + \frac{3}{2}\right)^2 + \frac{7}{4} \]