Solve each of the following equatins :
`2x^(2)-5x+3=0`

Answer

To solve the quadratic equation \(2x^2 - 5x + 3 = 0\), we can use the method of factoring. Here are the steps to solve the equation:

### Step 1: Identify the coefficients
In the equation \(2x^2 - 5x + 3 = 0\), we identify the coefficients:
- \(a = 2\)
- \(b = -5\)
- \(c = 3\)

### Step 2: Multiply \(a\) and \(c\)
Next, we multiply \(a\) and \(c\):
\[ ac = 2 \times 3 = 6 \]

### 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 to \(-5\) (the value of \(b\)). The numbers that satisfy this condition are \(-2\) and \(-3\) because:
\[ -2 \times -3 = 6 \quad \text{and} \quad -2 + -3 = -5 \]

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

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

### Step 6: Factor by grouping
Now we factor out the common terms in each group:
\[ 2x(x - 1) - 3(x - 1) = 0 \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \((x - 1)\):
\[ (2x - 3)(x - 1) = 0 \]

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

### Step 9: Solve for \(x\)
Now we solve each equation:
1. From \(2x - 3 = 0\):
\[ 2x = 3 \implies x = \frac{3}{2} \]

2. From \(x - 1 = 0\):
\[ x = 1 \]

### Step 10: Write the solutions
Thus, the solutions to the equation \(2x^2 - 5x + 3 = 0\) are:
\[ x = \frac{3}{2} \quad \text{and} \quad x = 1 \]

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