Find the values of x for the given equation `3x^(2) + 5x-2=0`

Options

A. `-3` and 2
B. 2 and `-3`
C. 3 and `-1/2`
D. `-2` and `1/3`

Answer

To solve the quadratic equation \(3x^2 + 5x - 2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 3\)
- \(b = 5\)
- \(c = -2\)

### Step 2: Calculate the product \(ac\)
We need to calculate the product of \(a\) and \(c\):
\[ ac = 3 \times (-2) = -6 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-6\) (the value of \(ac\)) and add up to \(5\) (the value of \(b\)). The numbers that satisfy this condition are \(6\) and \(-1\) because:
\[ 6 \times (-1) = -6 \quad \text{and} \quad 6 + (-1) = 5 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the equation by splitting the middle term \(5x\) into \(6x - 1x\):
\[ 3x^2 + 6x - 1x - 2 = 0 \]

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

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

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

### Step 8: Solve for \(x\)
1. For \(3x - 1 = 0\):
\[ 3x = 1 \implies x = \frac{1}{3} \]
2. For \(x + 2 = 0\):
\[ x = -2 \]

### Final Solution
The values of \(x\) that satisfy the equation \(3x^2 + 5x - 2 = 0\) are:
\[ x = \frac{1}{3} \quad \text{and} \quad x = -2 \]

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