https://file.aiquickdraw.com/mathbot/file/1727505687113tky1kxae.jpg

The problem provided in the image is written in Gujarati and reads as follows:

**Problem:**
"સમીકરણ \( x^2 - 3 \) ના શૂન્ય શોધો અને તેનું શૂન્ય અને ગુણાંકો વચ્ચેનો સંબંધ બતાવો."

**Translation:**
"Find the roots of the equation \( x^2 - 3 = 0 \), and show the relationship between its roots and coefficients."

### Step-by-Step Solution:

1. **Solve the equation \( x^2 - 3 = 0 \):**

   \[
   x^2 = 3
   \]

   Taking the square root on both sides:

   \[
   x = \pm \sqrt{3}
   \]

   So, the roots are:
   \[
   x_1 = \sqrt{3}, \quad x_2 = -\sqrt{3}
   \]

2. **Relationship between roots and coefficients:**

   For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the relationships between the roots \( \alpha \) and \( \beta \), and the coefficients are as follows:
   
   - Sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
   - Product of the roots: \( \alpha \cdot \beta = \frac{c}{a} \)
   
   In the given equation \( x^2 - 3 = 0 \), we have:
   
   \[
   a = 1, \quad b = 0, \quad c = -3
   \]
   
   So:
   - The sum of the roots \( \sqrt{3} + (-\sqrt{3}) = 0 \), which is consistent with \( -\frac{b}{a} = -\frac{0}{1} = 0 \).
   - The product of the roots \( \sqrt{3} \times (-\sqrt{3}) = -3 \), which is consistent with \( \frac{c}{a} = \frac{-3}{1} = -3 \).

Thus, the roots \( \sqrt{3} \) and \( -\sqrt{3} \) satisfy the equation, and their sum and product match the relationships between the roots and the coefficients.

---

Would you like any additional details or clarification? Here are some related questions for deeper understanding:

1. What are the steps to solve a general quadratic equation?
2. How does the discriminant help in determining the nature of the roots?
3. Can you explain how to derive the quadratic formula?
4. How would the solution change if the equation were \( x^2 - 3x + 2 = 0 \)?
5. How can we graphically represent the roots of a quadratic equation?

**Tip:** Always check if a quadratic equation can be factored before applying the quadratic formula for easier computation.

સમીકરણ x^2 - 3 ના શૂન્ય શોધો અને તેનું શૂન્ય અને ગુણાંકો વચ્ચેનો સંબંધ બતાવો.

Learn how to solve the quadratic equation x^2 - 3 = 0 and understand the relationship between the roots and coefficients. Suitable for students in Grades 7-9, this example covers key algebraic concepts and root-coefficient relationships.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation