https://file.aiquickdraw.com/mathbot/file/17261640471505jju325e.jpg

The exercise asks to find the "forme canonique" (vertex form) of the given quadratic functions. To convert these functions into their canonical form, we complete the square.

The general quadratic equation is:
\[
f(x) = ax^2 + bx + c
\]
The canonical form is:
\[
f(x) = a(x-h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.

### For part a: \( f(x) = 2x^2 - 5x + 10 \)

1. Factor out the coefficient of \(x^2\) from the first two terms:
   \[
   f(x) = 2(x^2 - \frac{5}{2}x) + 10
   \]

2. Complete the square:
   Take half of \(\frac{5}{2}\), square it, and add and subtract inside the parentheses:
   \[
   f(x) = 2\left(x^2 - \frac{5}{2}x + \left(\frac{5}{4}\right)^2 - \left(\frac{5}{4}\right)^2\right) + 10
   \]
   \[
   f(x) = 2\left(\left(x - \frac{5}{4}\right)^2 - \frac{25}{16}\right) + 10
   \]

3. Simplify the expression:
   \[
   f(x) = 2\left(x - \frac{5}{4}\right)^2 - \frac{25}{8} + 10
   \]
   \[
   f(x) = 2\left(x - \frac{5}{4}\right)^2 + \frac{55}{8}
   \]

Thus, the canonical form is:
\[
f(x) = 2\left(x - \frac{5}{4}\right)^2 + \frac{55}{8}
\]

### For part b: \( f(x) = x^2 - 3x + 1 \)

1. Complete the square:
   Take half of \(-3\), square it, and add and subtract inside the parentheses:
   \[
   f(x) = (x^2 - 3x + \left(\frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2) + 1
   \]
   \[
   f(x) = \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} + 1
   \]

2. Simplify the expression:
   \[
   f(x) = \left(x - \frac{3}{2}\right)^2 - \frac{5}{4}
   \]

Thus, the canonical form is:
\[
f(x) = \left(x - \frac{3}{2}\right)^2 - \frac{5}{4}
\]

Let me know if you need any clarification or further details. Here are 5 related questions:

1. How do you find the vertex from a quadratic equation in standard form?
2. What is the geometric interpretation of the canonical form?
3. Can all quadratic functions be converted to canonical form?
4. How does the value of \(a\) in the quadratic function affect the parabola's shape?
5. What is the difference between the standard form and canonical form of a quadratic function?

**Tip:** Completing the square is a useful technique not just for finding the vertex, but also for solving quadratic equations and integration problems in calculus.

Donner la forme canonique des fonctions suivantes : f(x) = 2x^2 - 5x + 10, f(x) = x^2 - 3x + 1

Learn how to find the canonical (vertex) form of the quadratic functions f(x) = 2x^2 - 5x + 10 and f(x) = x^2 - 3x + 1 by completing the square. This algebra exercise focuses on quadratic equations and is suitable for high school 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