Solve the following quations by using qardratic formula:
`x^(2)-18x+77=0`

Answer

To solve the quadratic equation \(x^2 - 18x + 77 = 0\) using the quadratic formula, we will follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, we can identify:
- \(a = 1\)
- \(b = -18\)
- \(c = 77\)

### 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 \(c\) into the formula
Substituting the values we identified:
\[ x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 77}}{2 \cdot 1} \]

### Step 4: Simplify the expression
Calculating the values:
\[ x = \frac{18 \pm \sqrt{324 - 308}}{2} \]
\[ x = \frac{18 \pm \sqrt{16}}{2} \]

### Step 5: Calculate the square root
The square root of 16 is 4:
\[ x = \frac{18 \pm 4}{2} \]

### Step 6: Solve for the two possible values of \(x\)
Now, we will calculate the two possible solutions:
1. \(x = \frac{18 + 4}{2} = \frac{22}{2} = 11\)
2. \(x = \frac{18 - 4}{2} = \frac{14}{2} = 7\)

### Final Answer
The solutions to the equation \(x^2 - 18x + 77 = 0\) are:
\[ x = 11 \quad \text{and} \quad x = 7 \]

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