Let's solve these two questions based on the images you provided.

Question 4

We are given the quadratic equation in the general form: $f(x) = -5x^2 + 20x - 3$ We need to rewrite it in standard form, which looks like $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

1. Complete the square: $f(x) = -5(x^2 - 4x) - 3$ To complete the square inside the parentheses, take half of $-4$ (which is $-2$) and square it to get $4$. So: $f(x) = -5((x^2 - 4x + 4) - 4) - 3$ $f(x) = -5((x - 2)^2 - 4) - 3$ Now distribute the $-5$: $f(x) = -5(x - 2)^2 + 20 - 3$ $f(x) = -5(x - 2)^2 + 17$

The standard form is $f(x) = -5(x - 2)^2 + 17$.

Since the coefficient of $(x - 2)^2$ is negative ($-5$), the parabola opens down.

So, the correct answer is:

• (-5(x - 2)^2 + 17; Opens Down.

Question 5

We are given the quadratic equation in the standard form: $f(x) = 10(x + 7)^2 - 35$ We need to rewrite this in the general form, which is $f(x) = ax^2 + bx + c$.

1. Expand the equation: $f(x) = 10(x^2 + 14x + 49) - 35$ Distribute the $10$: $f(x) = 10x^2 + 140x + 490 - 35$ Simplify: $f(x) = 10x^2 + 140x + 455$

The general form is $f(x) = 10x^2 + 140x + 455$.

Since the coefficient of $x^2$ is positive ($10$), the parabola opens up.

So, the correct answer is:

• (10x^2 + 140x + 455; Opens Up.

Would you like any additional details or have questions about these solutions?

Here are five related questions:

1. What is the process of completing the square, and why is it used?
2. How can you determine if a parabola opens up or down by just looking at the equation?
3. How do you find the vertex of a parabola from its standard form?
4. What are the differences between general and standard forms of quadratic equations?
5. Can you derive the vertex formula from the general form $ax^2 + bx + c$?

Tip: Always check the sign of the leading coefficient to know if the parabola opens up or down (positive means up, negative means down).