To solve the quadratic equation \( x^2 + 7x + 3 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \( a = 1 \), \( b = 7 \), and \( c = 3 \). Now, let's plug in these values:

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(1)(3)}}{2(1)}
\]

First, calculate the discriminant:

\[
\Delta = 7^2 - 4(1)(3) = 49 - 12 = 37
\]

Now substitute this into the formula:

\[
x = \frac{-7 \pm \sqrt{37}}{2}
\]

Thus, the two solutions are:

\[
x_1 = \frac{-7 + \sqrt{37}}{2}, \quad x_2 = \frac{-7 - \sqrt{37}}{2}
\]

These are the exact solutions. You can leave the answer like this or approximate it:

\[
x_1 \approx -0.46, \quad x_2 \approx -6.54
\]

Would you like more details on this solution or further clarification?

### Related questions:
1. What does the discriminant tell us about the nature of the roots?
2. How do you complete the square to solve a quadratic equation?
3. Can all quadratic equations be solved using factoring?
4. How would you solve this equation graphically?
5. What are some real-world applications of quadratic equations?

**Tip:** The discriminant \( b^2 - 4ac \) can help you quickly determine whether a quadratic equation has real or complex solutions.

Learn how to solve the quadratic equation x^2 + 7x + 3 with a detailed step-by-step guide using the quadratic formula. This solution explains key algebraic concepts, including calculating the discriminant, and is ideal for students in Grades 9-10.

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