Full solution

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

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y=x^{2}+3 x+8

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Answer:

y=

Focus on $x^2$ and $x$ terms: To complete the square and rewrite the quadratic function in vertex form, we first need to focus on the $x^2$ and $x$ terms. The vertex form of a quadratic function is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. To complete the square, we need to create a perfect square trinomial from the $x^2$ and $x$ terms.
Create perfect square trinomial: The coefficient of $x^2$ is $1$ , which is already in the form we need. To create a perfect square trinomial, we take the coefficient of $x$ , which is $3$ , divide it by $2$ , and square the result. This will give us the constant term we need to add and then subtract to complete the square. $\newline$ $(\frac{3}{2})^2 = 2.25$ or $\frac{9}{4}$
Add and subtract constant term: We add and subtract $(\frac{3}{2})^2$ inside the equation to maintain the equality. We add it to create the perfect square trinomial and subtract it to keep the equation balanced. $\newline$ $y = x^2 + 3x + (\frac{3}{2})^2 - (\frac{3}{2})^2 + 8$
Simplify the equation: Now we simplify the equation by combining the constant terms. $\newline$ $y = x^2 + 3x + \frac{9}{4} - \frac{9}{4} + 8$ $\newline$ $y = x^2 + 3x + \frac{9}{4} + \frac{32}{4} - \frac{9}{4}$ $\newline$ $y = x^2 + 3x + \frac{23}{4}$
Factor perfect square trinomial: Next, we factor the perfect square trinomial and combine the constant terms outside the square. $\newline$ $y = (x + \frac{3}{2})^2 + \frac{23}{4}$
Quadratic function in vertex form: This is the quadratic function in vertex form. The vertex form is $y = a(x - h)^2 + k$ , so our $h$ is $-\frac{3}{2}$ and our $k$ is $\frac{23}{4}$ .