What are the roots of the quadratic equation `2x^(2) - 7 x + 6 =0` ?

Options

A. `2, - (3)/(2)`
B. `2, ( 3)/(2)`
C. `2, (2)/(3)`
D. `-2, (3)/(2)`

Answer

To find the roots of the quadratic equation \(2x^2 - 7x + 6 = 0\), we can use the middle term splitting method. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 2\)
- \(b = -7\)
- \(c = 6\)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply \(a\) and \(c\):
\[ a \cdot c = 2 \cdot 6 = 12 \]

### Step 3: Find two numbers that multiply to \(12\) and add to \(-7\)
We need to find two numbers that multiply to \(12\) (the product from Step 2) and add up to \(-7\) (the coefficient \(b\)). The two numbers that satisfy these conditions are \(-3\) and \(-4\) because:
\[ -3 \cdot -4 = 12 \quad \text{and} \quad -3 + -4 = -7 \]

### Step 4: Rewrite the middle term
Now we can rewrite the equation by splitting the middle term using the numbers found:
\[ 2x^2 - 3x - 4x + 6 = 0 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (2x^2 - 3x) + (-4x + 6) = 0 \]

### Step 6: Factor by grouping
Now, we factor out the common factors from each group:
\[ x(2x - 3) - 2(2x - 3) = 0 \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \((2x - 3)\):
\[ (2x - 3)(x - 2) = 0 \]

### Step 8: Set each factor to zero
Now, we set each factor equal to zero to find the roots:
1. \(2x - 3 = 0\)
2. \(x - 2 = 0\)

### Step 9: Solve for \(x\)
Solving these equations gives:
1. For \(2x - 3 = 0\):
\[ 2x = 3 \implies x = \frac{3}{2} \]
2. For \(x - 2 = 0\):
\[ x = 2 \]

### Conclusion
The roots of the quadratic equation \(2x^2 - 7x + 6 = 0\) are:
\[ x = \frac{3}{2} \quad \text{and} \quad x = 2 \]