I. ` 12x^(2) + 17x+ 6 = 0" " II. 6y^(2) + 5y + 1 = 0`

Options

A. ` x lt y`
B. `x gt y`
C. ` x le y`
D. `x ge y`

Explanation

To solve the given quadratic equations step by step, we will use the factorization method.

### Step 1: Solve the first equation \(12x^2 + 17x + 6 = 0\)

1. **Identify the coefficients**:
- \(a = 12\)
- \(b = 17\)
- \(c = 6\)

2. **Multiply \(a\) and \(c\)**:
- \(ac = 12 \times 6 = 72\)

3. **Find two numbers that multiply to \(ac\) (72) and add to \(b\) (17)**:
- The numbers are \(8\) and \(9\) because \(8 \times 9 = 72\) and \(8 + 9 = 17\).

4. **Rewrite the equation using these numbers**:
- \(12x^2 + 8x + 9x + 6 = 0\)

5. **Group the terms**:
- \((12x^2 + 8x) + (9x + 6) = 0\)

6. **Factor by grouping**:
- \(4x(3x + 2) + 3(3x + 2) = 0\)
- \((4x + 3)(3x + 2) = 0\)

7. **Set each factor to zero**:
- \(4x + 3 = 0\) or \(3x + 2 = 0\)

8. **Solve for \(x\)**:
- From \(4x + 3 = 0\):
\[ 4x = -3 \implies x = -\frac{3}{4} \]
- From \(3x + 2 = 0\):
\[ 3x = -2 \implies x = -\frac{2}{3} \]

### Step 2: Solve the second equation \(6y^2 + 5y + 1 = 0\)

1. **Identify the coefficients**:
- \(a = 6\)
- \(b = 5\)
- \(c = 1\)

2. **Multiply \(a\) and \(c\)**:
- \(ac = 6 \times 1 = 6\)

3. **Find two numbers that multiply to \(ac\) (6) and add to \(b\) (5)**:
- The numbers are \(2\) and \(3\) because \(2 \times 3 = 6\) and \(2 + 3 = 5\).

4. **Rewrite the equation using these numbers**:
- \(6y^2 + 2y + 3y + 1 = 0\)

5. **Group the terms**:
- \((6y^2 + 2y) + (3y + 1) = 0\)

6. **Factor by grouping**:
- \(2y(3y + 1) + 1(3y + 1) = 0\)
- \((2y + 1)(3y + 1) = 0\)

7. **Set each factor to zero**:
- \(2y + 1 = 0\) or \(3y + 1 = 0\)

8. **Solve for \(y\)**:
- From \(2y + 1 = 0\):
\[ 2y = -1 \implies y = -\frac{1}{2} \]
- From \(3y + 1 = 0\):
\[ 3y = -1 \implies y = -\frac{1}{3} \]

### Summary of Solutions:
- The values of \(x\) are \(x = -\frac{3}{4}\) and \(x = -\frac{2}{3}\).
- The values of \(y\) are \(y = -\frac{1}{2}\) and \(y = -\frac{1}{3}\).

### Step 3: Compare \(x\) and \(y\)

1. **Compare the values**:
- \(x = -\frac{3}{4} \approx -0.75\)
- \(x = -\frac{2}{3} \approx -0.67\)
- \(y = -\frac{1}{2} = -0.5\)
- \(y = -\frac{1}{3} \approx -0.33\)

2. **Determine the relationship**:
- Since both values of \(x\) are less than both values of \(y\), we conclude:
- \(x < y\)

### Final Answer:
The relationship between \(x\) and \(y\) is \(x < y\).

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