To solve the quadratic equation $x^2 + 3x - 10 = 0$ by completing the square, follow these steps:

Step 1: Move the constant term to the right side

Start with the given equation: $x^2 + 3x - 10 = 0$ Add 10 to both sides: $x^2 + 3x = 10$

Step 2: Complete the square

To complete the square, take the coefficient of $x$ (which is 3), divide it by 2, and then square it: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$ Add $\frac{9}{4}$ to both sides of the equation: $x^2 + 3x + \frac{9}{4} = 10 + \frac{9}{4}$

Step 3: Simplify the equation

On the left side, you now have a perfect square trinomial: $\left(x + \frac{3}{2}\right)^2 = \frac{40}{4} + \frac{9}{4}$ Simplify the right side: $\left(x + \frac{3}{2}\right)^2 = \frac{49}{4}$

Step 4: Solve for $x$

Take the square root of both sides: $x + \frac{3}{2} = \pm \frac{7}{2}$

Step 5: Isolate $x$

Solve for $x$ by subtracting $\frac{3}{2}$ from both sides:

1. For the positive square root: $x = \frac{7}{2} - \frac{3}{2} = \frac{4}{2} = 2$
2. For the negative square root: $x = -\frac{7}{2} - \frac{3}{2} = -\frac{10}{2} = -5$

Final Answer:

The solutions are $x = 2$ and $x = -5$.

So, the answer is: $2, -5$

Would you like a deeper breakdown of any of these steps, or do you have more questions?

Here are 5 relative questions:

1. How do you solve quadratic equations by factoring?
2. What is the quadratic formula and how does it work?
3. Can you derive the quadratic formula from completing the square?
4. What are real-life applications of quadratic equations?
5. How does the discriminant determine the nature of the roots?

Tip: Completing the square is particularly useful for deriving the quadratic formula and understanding the symmetry of parabolas!