Solve the quadratic equation x2+3x−10=0 by completing the square.

What is the solution, or what are the solutions, to the equation?

Enter your answer as the correct value or values. If there is more than one solution, use a comma between the values, like this: 42, 53




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!

Learn how to solve the quadratic equation x^2 + 3x - 10 by completing the square with a detailed step-by-step solution. This problem covers key algebraic concepts, including completing the square and the quadratic formula. Suitable for students in Grades 9-12.

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