https://file.aiquickdraw.com/mathbot/file/17277309179750zwrcgtq.jpg

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).

Change from the general form f(x) = -5x^2 + 20x - 3 to the standard form and determine if the parabola opens up or down. Change from the standard form f(x) = 10(x + 7)^2 - 35 to the general form and determine if the parabola opens up or down.

Learn how to convert quadratic equations from general form to standard form and vice versa. Understand how to determine whether the parabola opens up or down based on the coefficient of x^2. This is a common topic in high school algebra, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation