Find the value of X: `8x^(2)+10x+3=0`

Answer

To solve the quadratic equation \(8x^2 + 10x + 3 = 0\), we can use the method of factorization. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \(ax^2 + bx + c = 0\), where:
- \(a = 8\)
- \(b = 10\)
- \(c = 3\)

### Step 2: Calculate \(a \cdot c\)
We need to find the product of \(a\) and \(c\):
\[ a \cdot c = 8 \cdot 3 = 24 \]

### Step 3: Find two numbers that multiply to \(24\) and add to \(10\)
We need to find two numbers that multiply to \(24\) (the product from Step 2) and add up to \(10\) (the coefficient \(b\)). The numbers that satisfy this condition are \(4\) and \(6\) because:
\[ 4 \cdot 6 = 24 \quad \text{and} \quad 4 + 6 = 10 \]

### Step 4: Rewrite the middle term
Using the two numbers found, we can rewrite the equation:
\[ 8x^2 + 4x + 6x + 3 = 0 \]

### Step 5: Factor by grouping
Now, we can group the terms:
\[ (8x^2 + 4x) + (6x + 3) = 0 \]
Factoring out the common factors in each group:
\[ 4x(2x + 1) + 3(2x + 1) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((2x + 1)\):
\[ (4x + 3)(2x + 1) = 0 \]

### Step 7: Set each factor to zero
Now, we set each factor equal to zero:
1. \(4x + 3 = 0\)
2. \(2x + 1 = 0\)

### Step 8: Solve for \(x\)
For the first equation:
\[ 4x + 3 = 0 \implies 4x = -3 \implies x = -\frac{3}{4} \]

For the second equation:
\[ 2x + 1 = 0 \implies 2x = -1 \implies x = -\frac{1}{2} \]

### Final Solution
The values of \(x\) are:
\[ x = -\frac{3}{4} \quad \text{or} \quad x = -\frac{1}{2} \]

---