To solve the equation \((2x + 3)^2 + (x + 4)^2 = 10\), follow these steps:

### Step 1: Expand both squares
\[
(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9
\]
\[
(x + 4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16
\]

### Step 2: Add the expanded terms
\[
(4x^2 + 12x + 9) + (x^2 + 8x + 16) = 10
\]
\[
4x^2 + x^2 + 12x + 8x + 9 + 16 = 10
\]
\[
5x^2 + 20x + 25 = 10
\]

### Step 3: Simplify the equation
Subtract 10 from both sides:
\[
5x^2 + 20x + 25 - 10 = 0
\]
\[
5x^2 + 20x + 15 = 0
\]

### Step 4: Divide the equation by 5
\[
x^2 + 4x + 3 = 0
\]

### Step 5: Solve the quadratic equation
Factor the quadratic equation:
\[
(x + 1)(x + 3) = 0
\]

### Step 6: Find the solutions
Set each factor equal to zero:
\[
x + 1 = 0 \quad \text{or} \quad x + 3 = 0
\]
\[
x = -1 \quad \text{or} \quad x = -3
\]

Thus, the solutions are \(x = -1\) and \(x = -3\).

Would you like more details or have any questions?

---

Here are 5 related questions for further practice:
1. Solve \((3x + 2)^2 + (x - 1)^2 = 25\).
2. Solve the quadratic equation \(2x^2 + 7x + 3 = 0\).
3. Factor the quadratic expression \(x^2 + 5x + 6 = 0\).
4. Find the roots of \(x^2 + 2x - 8 = 0\).
5. Solve \((x + 5)^2 + (2x + 1)^2 = 13\).

**Tip:** When solving quadratic equations, always check if they can be factored before resorting to the quadratic formula. It saves time!

Discover how to solve the equation (2x + 3)² + (x + 4)² = 10 with a detailed, step-by-step guide. This tutorial covers expanding binomials, combining like terms, and factoring quadratic equations, making it perfect for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation