https://file.aiquickdraw.com/mathbot/file/1728997683607lak0dz7c.jpg

solve this equation by writing this in vertex form

The given function is:

\[
f(x) = \frac{3x^2 + 3x - 12}{3}
\]

First, simplify the equation by dividing every term inside the parentheses by 3:

\[
f(x) = x^2 + x - 4
\]

Now, to write this in vertex form, we will complete the square.

1. Start with the equation:
\[
f(x) = x^2 + x - 4
\]

2. Focus on the quadratic and linear terms: \(x^2 + x\). To complete the square, take half of the coefficient of \(x\), square it, and add/subtract it inside the equation.

The coefficient of \(x\) is 1. Half of 1 is \( \frac{1}{2} \), and squaring it gives \( \frac{1}{4} \).

3. Add and subtract \( \frac{1}{4} \) inside the equation:

\[
f(x) = (x^2 + x + \frac{1}{4}) - \frac{1}{4} - 4
\]

4. Now, rewrite the trinomial \(x^2 + x + \frac{1}{4}\) as a perfect square:

\[
f(x) = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} - 4
\]

5. Combine constants \( - \frac{1}{4} \) and \( -4 \):

\[
f(x) = \left(x + \frac{1}{2}\right)^2 - \frac{17}{4}
\]

Thus, the vertex form of the equation is:

\[
f(x) = \left(x + \frac{1}{2}\right)^2 - \frac{17}{4}
\]

Let me know if you'd like further explanation or have any questions.

### Relative Questions:
1. What is the vertex of this quadratic function?
2. How do you find the axis of symmetry from the vertex form?
3. Can you solve for the x-intercepts using the vertex form?
4. What is the significance of the constant term in the vertex form?
5. How does completing the square help in solving quadratic equations?

### Tip:
In vertex form, the coordinates of the vertex \((h, k)\) are easily identified as \((-h, k)\) from the expression \(f(x) = a(x-h)^2 + k\).

Solve the equation by writing it in vertex form.

Learn how to convert the quadratic function f(x) = (3x^2 + 3x - 12)/3 into its vertex form using the completing the square method. This detailed guide covers algebraic concepts suitable for students in 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