Complete the square to re-write the quadratic function in vertex form:

y, equals, x, squared, plus, 3, x, plus, 4
y=x
2
+3x+4

1 answer

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.

  1. Start with the original function: \[ y = x^2 + 3x + 4 \]

  2. 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 \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \)).

  3. Rewrite the equation by adding and subtracting \( \frac{9}{4} \): \[ y = x^2 + 3x + \frac{9}{4} - \frac{9}{4} + 4 \]

  4. This can be reorganized as: \[ y = \left(x^2 + 3x + \frac{9}{4}\right) + 4 - \frac{9}{4} \]

  5. The expression in the parentheses is a perfect square: \[ y = \left(x + \frac{3}{2}\right)^2 + 4 - \frac{9}{4} \]

  6. Now combine \( 4 \) and \( -\frac{9}{4} \). Convert 4 to fractions with the same denominator: \[ 4 = \frac{16}{4} \]

    Thus, \[ 4 - \frac{9}{4} = \frac{16}{4} - \frac{9}{4} = \frac{7}{4} \]

  7. 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: \[ y = \left(x + \frac{3}{2}\right)^2 + \frac{7}{4} \]