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The equation you provided is:

\[
x^2 - 3x - 7 = 0
\]

This is a quadratic equation, and we can solve it using the quadratic formula:

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

For the equation \(x^2 - 3x - 7 = 0\), we have:
- \(a = 1\)
- \(b = -3\)
- \(c = -7\)

Now let's calculate the discriminant:

\[
\Delta = b^2 - 4ac = (-3)^2 - 4(1)(-7) = 9 + 28 = 37
\]

Next, applying the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{37}}{2(1)} = \frac{3 \pm \sqrt{37}}{2}
\]

Thus, the two roots are:

\[
x_1 = \frac{3 + \sqrt{37}}{2}, \quad x_2 = \frac{3 - \sqrt{37}}{2}
\]

Since we are asked for the smaller root, we take the root with the negative sign:

\[
x_2 = \frac{3 - \sqrt{37}}{2}
\]

The approximate value of the smaller root is:

\[
x_2 \approx \frac{3 - 6.08}{2} = \frac{-3.08}{2} \approx -1.54
\]

Therefore, the smaller root is approximately \(-1.54\).

Would you like more details or have any questions?

Here are some follow-up questions to deepen understanding:

1. What is the significance of the discriminant in a quadratic equation?
2. Can you calculate the larger root of this quadratic equation?
3. How would you solve this equation graphically?
4. What happens if the discriminant is negative?
5. How do the roots of the quadratic equation relate to its graph?

**Tip**: Always check the sign of the discriminant before solving to predict the nature of the roots (real or complex).

Resuelve e indica la menor raíz: x^2 - 3x - 7 = 0

Learn how to solve the quadratic equation x^2 - 3x - 7 and find the smallest root. This step-by-step guide covers key algebraic concepts using the quadratic formula and is suitable for students in Grades 8-10.

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