Solve `5x^(2) + 11x + 6= 0`

Answer

To solve the quadratic equation \(5x^2 + 11x + 6 = 0\), we can use the method of factoring. Here are the steps to solve the equation:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 5\)
- \(b = 11\)
- \(c = 6\)

### Step 2: Calculate the product \(ac\)
We need to calculate the product of \(a\) and \(c\):
\[ ac = 5 \times 6 = 30 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(30\) (the value of \(ac\)) and add up to \(11\) (the value of \(b\)). The numbers \(5\) and \(6\) satisfy this condition:
\[ 5 \times 6 = 30 \quad \text{and} \quad 5 + 6 = 11 \]

### Step 4: Rewrite the middle term using the two numbers
Now, we can rewrite the equation by breaking the middle term \(11x\) into \(5x + 6x\):
\[ 5x^2 + 5x + 6x + 6 = 0 \]

### Step 5: Factor by grouping
Next, we group the terms:
\[ (5x^2 + 5x) + (6x + 6) = 0 \]
Factoring out the common terms in each group gives us:
\[ 5x(x + 1) + 6(x + 1) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((x + 1)\):
\[ (5x + 6)(x + 1) = 0 \]

### Step 7: Set each factor to zero
We set each factor equal to zero:
1. \(5x + 6 = 0\)
2. \(x + 1 = 0\)

### Step 8: Solve for \(x\)
For the first equation:
\[ 5x + 6 = 0 \implies 5x = -6 \implies x = -\frac{6}{5} \]
For the second equation:
\[ x + 1 = 0 \implies x = -1 \]

### Final Solutions
The solutions to the equation \(5x^2 + 11x + 6 = 0\) are:
\[ x = -1 \quad \text{and} \quad x = -\frac{6}{5} \]

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