Solve the following quations by using qardratic formula:
`2x^(2)-9x+7=0`

Answer

To solve the quadratic equation \(2x^2 - 9x + 7 = 0\) using the quadratic formula, we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). From the given equation, we can identify:
- \(a = 2\)
- \(b = -9\)
- \(c = 7\)

### Step 2: Write the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Substitute the values of \(a\), \(b\), and \(c\)
Substituting the values we identified:
\[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 2 \cdot 7}}{2 \cdot 2} \]

### Step 4: Simplify the expression
Calculating \( -(-9) \):
\[ x = \frac{9 \pm \sqrt{81 - 56}}{4} \]

Calculating \(81 - 56\):
\[ x = \frac{9 \pm \sqrt{25}}{4} \]

### Step 5: Calculate the square root
The square root of \(25\) is \(5\):
\[ x = \frac{9 \pm 5}{4} \]

### Step 6: Find the two possible values for \(x\)
Now we will calculate the two possible values for \(x\):
1. \(x = \frac{9 + 5}{4} = \frac{14}{4} = \frac{7}{2}\)
2. \(x = \frac{9 - 5}{4} = \frac{4}{4} = 1\)

### Final Answer
The solutions to the equation \(2x^2 - 9x + 7 = 0\) are:
\[ x = \frac{7}{2} \quad \text{and} \quad x = 1 \]

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