Which of the following equatiosn has the sum of its roots as 7?

Options

A. `x ^(2) - 7x + 14 =0`
B. `x ^(2) - 7 x + 10 =0`
C. `x ^(2) + 7 x + 6=0`
D. `x ^(2) + 5x -6=0`

Answer

To determine which of the given equations has a sum of roots equal to 7, we can use the property of quadratic equations. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the sum of the roots can be calculated using the formula:

\[ \text{Sum of roots} = -\frac{b}{a} \]

Let's analyze the given options step by step.

### Step 1: Analyze the first equation
Assume the first equation is \( x^2 - 7x + 14 = 0 \).

- Here, \( a = 1 \) and \( b = -7 \).
- Calculate the sum of the roots:
\[ \text{Sum of roots} = -\frac{-7}{1} = 7 \]
- This equation has a sum of roots equal to 7.

### Step 2: Analyze the second equation
Assume the second equation is \( x^2 - 7x + 10 = 0 \).

- Here, \( a = 1 \) and \( b = -7 \).
- Calculate the sum of the roots:
\[ \text{Sum of roots} = -\frac{-7}{1} = 7 \]
- This equation also has a sum of roots equal to 7.

### Step 3: Check for real roots
To ensure that both equations have real roots, we can check the discriminant (\( D \)):

For \( x^2 - 7x + 14 = 0 \):
\[ D = b^2 - 4ac = (-7)^2 - 4(1)(14) = 49 - 56 = -7 \]
Since \( D < 0 \), this equation does not have real roots.

For \( x^2 - 7x + 10 = 0 \):
\[ D = b^2 - 4ac = (-7)^2 - 4(1)(10) = 49 - 40 = 9 \]
Since \( D > 0 \), this equation has real roots.

### Conclusion
Only the second equation \( x^2 - 7x + 10 = 0 \) has real roots and a sum of roots equal to 7.

### Final Answer
The equation that has the sum of its roots as 7 is \( x^2 - 7x + 10 = 0 \).