Using the quadratic formula, solve the quadratic equation: x^2– 9x + 14 = 0

Options

A. 2 and 7
B. - 2 and 7
C. 2 and -7
D. -2 and - 7

Answer

To solve the quadratic equation \( x^2 - 9x + 14 = 0 \) using the quadratic formula, we will follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In our equation:
- \( a = 1 \)
- \( b = -9 \)
- \( c = 14 \)

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

### Step 3: Calculate the discriminant
First, we need to calculate the discriminant \( D \):
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-9)^2 - 4 \cdot 1 \cdot 14 = 81 - 56 = 25 \]

### Step 4: Substitute values into the quadratic formula
Now we can substitute \( b \) and \( D \) into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{25}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{9 \pm 5}{2} \]

### Step 5: Calculate the two possible values for \( x \)
Now we will calculate the two possible values for \( x \):
1. For the positive case:
\[ x = \frac{9 + 5}{2} = \frac{14}{2} = 7 \]
2. For the negative case:
\[ x = \frac{9 - 5}{2} = \frac{4}{2} = 2 \]

### Step 6: Write the final solutions
The solutions to the equation \( x^2 - 9x + 14 = 0 \) are:
\[ x = 2 \quad \text{and} \quad x = 7 \]