We are tasked with finding the real root of the quadratic equation:

$x^2 - 4x - 10 = 0$

Step 1: Solve using the quadratic formula

The quadratic formula is given by:

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

For the equation $x^2 - 4x - 10 = 0$, the coefficients are:

• $a = 1$
• $b = -4$
• $c = -10$

Substitute these values into the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-10)}}{2(1)}$

$x = \frac{4 \pm \sqrt{16 + 40}}{2}$

$x = \frac{4 \pm \sqrt{56}}{2}$

$x = \frac{4 \pm \sqrt{4 \times 14}}{2}$

$x = \frac{4 \pm 2\sqrt{14}}{2}$

Simplify the expression:

$x = 2 \pm \sqrt{14}$

Step 2: Approximate the root

Now we calculate the approximate value of $\sqrt{14}$:

$\sqrt{14} \approx 3.742$

Thus, the roots are:

$x = 2 + 3.742 \quad \text{or} \quad x = 2 - 3.742$

These give:

$x \approx 5.742 \quad \text{or} \quad x \approx -1.742$

Step 3: Conclusion

The real roots of the equation $x^2 - 4x - 10 = 0$ are approximately:

$x \approx 5.742 \quad \text{and} \quad x \approx -1.742$

Thus, the real root correct up to three decimal places is 5.742.

Would you like any further clarification or details on the steps?

Additional Questions:

1. How can you use the quadratic formula for equations with complex roots?
2. What is the significance of the discriminant in the quadratic formula?
3. Can you solve the quadratic equation using factoring? If yes, how?
4. How do you determine if a quadratic equation has real or complex roots?
5. What is the effect of changing the coefficients in the quadratic equation on the nature of its roots?

Tip: The discriminant ($b^2 - 4ac$) helps determine whether the quadratic equation has real or complex roots. If it's positive, the roots are real; if it's negative, the roots are complex.