The nature of the roots of the equation `x^(2)-5x+7=0` is

Options

A. no real roots
B. 1 real root
C. can't be determined
D. none of these

Answer

To determine the nature of the roots of the quadratic equation \( x^2 - 5x + 7 = 0 \), we will use the discriminant method. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\[ D = b^2 - 4ac \]

### Step 1: Identify the coefficients
For the equation \( x^2 - 5x + 7 = 0 \):
- \( a = 1 \)
- \( b = -5 \)
- \( c = 7 \)

### Step 2: Calculate the discriminant
Now, we will substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:

\[ D = (-5)^2 - 4 \cdot 1 \cdot 7 \]

Calculating \( (-5)^2 \):

\[ D = 25 - 4 \cdot 1 \cdot 7 \]

Calculating \( 4 \cdot 1 \cdot 7 \):

\[ D = 25 - 28 \]

Now, subtract:

\[ D = 25 - 28 = -3 \]

### Step 3: Determine the nature of the roots
The nature of the roots can be determined based on the value of the discriminant \( D \):
- If \( D > 0 \): The roots are real and distinct.
- If \( D = 0 \): The roots are real and equal.
- If \( D < 0 \): The roots are imaginary (or complex).

Since we found that \( D = -3 \), which is less than 0, we conclude that:

The roots of the equation \( x^2 - 5x + 7 = 0 \) are imaginary.

### Final Answer
The nature of the roots of the equation \( x^2 - 5x + 7 = 0 \) is **imaginary**.

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