Given the equation 4x2 − 8x + 20 = 0, what are the values of h and k when the equation is written in vertex form a(x − h)2 + k = 0? (1 point)

h = 4, k = −16

h = 4, k = −1

h = 1, k = −24

h = 1, k = 16

To rewrite the given quadratic equation \( 4x^2 - 8x + 20 = 0 \) in vertex form \( a(x - h)^2 + k = 0 \), follow these steps:

### Step 1: Factor out the coefficient of \( x^2 \) from the first two terms
\[
4x^2 - 8x + 20 = 0
\]
\[
4(x^2 - 2x) + 20 = 0
\]

### Step 2: Complete the square inside the parentheses
Take half of the coefficient of \( x \) (which is \(-2\)) and square it:
\[
\left(\frac{-2}{2}\right)^2 = 1
\]

Add and subtract this value inside the parentheses:
\[
4(x^2 - 2x + 1 - 1) + 20 = 0
\]
\[
4((x - 1)^2 - 1) + 20 = 0
\]

### Step 3: Simplify the equation
Distribute the \( 4 \):
\[
4(x - 1)^2 - 4 + 20 = 0
\]
\[
4(x - 1)^2 + 16 = 0
\]

### Step 4: Identify \( h \) and \( k \)
The vertex form of the equation is:
\[
a(x - h)^2 + k = 0
\]
Here, \( h = 1 \) and \( k = 16 \).

### Final Answer:
\[
h = 1, k = 16
\]

If you have further questions or want to explore the vertex form process more deeply, let me know! Here are some related questions:

1. What is the significance of the vertex in a parabola?
2. How do you determine the axis of symmetry from the vertex form?
3. Can the vertex form be used to determine the direction of the parabola?
4. What is the role of the coefficient \( a \) in the vertex form?
5. How can you convert a quadratic equation from vertex form back to standard form?

**Tip:** Completing the square is a powerful technique to rewrite quadratic equations in various forms—practice it to master quadratic transformations!

Given the equation 4x2 − 8x + 20 = 0, what are the values of h and k when the equation is written in vertex form a(x − h)^2 + k = 0? (1 point)
h = 4, k = −16
h = 4, k = −1
h = 1, k = −24
h = 1, k = 16

Explore how to rewrite the quadratic equation 4x^2 − 8x + 20 = 0 in vertex form and determine the values of h and k. This step-by-step solution covers completing the square and analyzing the vertex form of quadratic equations, ideal for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation