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The equation of a parabola is $y = 5x^2 + 20x + 18$ . Write the equation in vertex form. $\newline$ Write any numbers as integers or simplified proper or improper fractions. $\newline$ ______

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Write a quadratic function in vertex form

Full solution

Q. The equation of a parabola is

y = 5x^2 + 20x + 18

. Write the equation in vertex form.

\newline

Write any numbers as integers or simplified proper or improper fractions.

\newline

______

Given equation: Start with the given quadratic equation. $\newline$ Given equation: $y = 5x^2 + 20x + 18$ $\newline$ We need to rewrite this equation in vertex form, which is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
Factor out $x$ terms: Factor out the coefficient of the $x^2$ term from the $x$ terms. $\newline$ Factor out $5$ from the $x$ terms in the equation. $\newline$ $y = 5(x^2 + 4x) + 18$
Complete the square: Complete the square for the expression inside the parentheses. $\newline$ To complete the square, we need to add and subtract $(b/2)^2$ , where $b$ is the coefficient of $x$ . $\newline$ For $x^2 + 4x$ , $b$ is $4$ . $\newline$ $(4/2)^2 = (2)^2 = 4$ $\newline$ Add and subtract $4$ inside the parentheses. $\newline$ $y = 5(x^2 + 4x + 4 - 4) + 18$
Rewrite as a perfect square: Rewrite the expression inside the parentheses as a perfect square. $\newline$ The expression $x^2 + 4x + 4$ is a perfect square and can be written as $(x + 2)^2$ . $\newline$ $y = 5((x + 2)^2 - 4) + 18$
Distribute and simplify: Distribute the $5$ and simplify the equation. $y = 5(x + 2)^2 - 5(4) + 18$ $y = 5(x + 2)^2 - 20 + 18$ Combine the constant terms. $y = 5(x + 2)^2 - 2$

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