What is the correct first step when completing the square for the expression $x^2 + 5y^2 - 10x + 20y - 1$?

Group the terms by variable: $(x^2 - 10x) + (5y^2 + 20y) - 1$.

Factor out 5 from the entire expression.

Add and subtract $(10/2)^2$ to the x-terms.

Treat x as a constant and work only with the y-terms.

Quadratic Optimization Practice — Grade 4 amc_math

Lesson 15.4: Quadratic Optimization — Practice Questions

1. After completing the square, the expression $x^2 + y^2 + 8x - 2y + 10$ can be written in the form $(x+4)^2 + (y-1)^2 + C$. What is the value of $C$? ___
2. Which expression is equivalent to $3x^2 + y^2 - 18x + 4y + 20$ after completing the square?
- A. $3(x-3)^2 + (y+2)^2 - 11$
- B. $3(x-3)^2 + (y+2)^2 + 1$
- C. $(x-3)^2 + (y+2)^2 - 11$
- D. $3(x-9)^2 + (y+2)^2 - 251$
3. The expression $x^2 + 2y^2 + 6x - 8y + 15$ is rewritten as $(x+3)^2 + 2(y-k)^2 - 2$. What is the value of $k$? ___
4. What is the correct first step when completing the square for the expression $x^2 + 5y^2 - 10x + 20y - 1$?
- A. Group the terms by variable: $(x^2 - 10x) + (5y^2 + 20y) - 1$.
- B. Factor out 5 from the entire expression.
- C. Add and subtract $(10/2)^2$ to the x-terms.
- D. Treat x as a constant and work only with the y-terms.
5. When $4x^2 + 9y^2 + 8x - 36y + 4$ is written in the form $4(x+1)^2 + 9(y-2)^2 + C$, what is the value of $C$? ___
6. A condo association's monthly dues are 120 dollars. For each new tenant, dues are reduced by 5 dollars. Write an expression for the dues if $x$ new tenants move in. The expression is ___.
7. A company's daily profit is modeled by $P(x) = -3x^2 + 180x - 600$, where $x$ is the price of an item. What price $x$ in dollars maximizes the profit? ___
8. Does the function $f(x) = 4x^2 - 8x + 15$ have a maximum or a minimum value?
- A. A maximum value, because the parabola opens upward.
- B. A minimum value, because the parabola opens upward.
- C. A maximum value, because the parabola opens downward.
- D. A minimum value, because the parabola opens downward.
9. The height of a thrown ball is given by $h(t) = -8t^2 + 32t + 2$, where $t$ is time in seconds. After how many seconds does the ball reach its maximum height? ___
10. A manufacturer's weekly revenue is given by the function $R(n) = n(600 - 15n)$, where $n$ is the number of units sold. How many units must be sold to maximize revenue? ___