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

Answer

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

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

### Step 2: Calculate the product \(ac\)
Next, we calculate the product of \(a\) and \(c\):
\[ ac = 2 \times 3 = 6 \]

### Step 3: Find two numbers that sum to \(b\) and multiply to \(ac\)
We need to find two numbers that add up to \(b\) (which is \(-5\)) and multiply to \(ac\) (which is \(6\)). The numbers that satisfy these conditions are \(-2\) and \(-3\) because:
\[ -2 + (-3) = -5 \quad \text{and} \quad -2 \times -3 = 6 \]

### Step 4: Rewrite the middle term
We can now rewrite the quadratic equation by breaking the middle term:
\[ 2x^2 - 2x - 3x + 3 = 0 \]

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

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

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

### Step 8: Solve for \(x\)
Now we solve each equation:
1. From \(2x - 3 = 0\):
\[ 2x = 3 \implies x = \frac{3}{2} \]
2. From \(x - 1 = 0\):
\[ x = 1 \]

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

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