If a + b = 5, ab = -50 then find the value of a - b.

Options

A. 10
B. 25
C. 15
D. -10

Answer

To find the value of \( a - b \) given that \( a + b = 5 \) and \( ab = -50 \), we can follow these steps:

### Step 1: Use the given equations
We have:
1. \( a + b = 5 \)
2. \( ab = -50 \)

### Step 2: Set up the equation for \( a \) and \( b \)
We can represent \( a \) and \( b \) as the roots of a quadratic equation. The standard form of a quadratic equation based on the sum and product of its roots is:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values we have:
\[ x^2 - 5x - 50 = 0 \]

### Step 3: Solve the quadratic equation
To solve for \( x \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -5 \), and \( c = -50 \). Plugging in these values:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-50)}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 + 200}}{2} \]
\[ x = \frac{5 \pm \sqrt{225}}{2} \]
\[ x = \frac{5 \pm 15}{2} \]

### Step 4: Calculate the roots
Calculating the two possible values for \( x \):
1. \( x = \frac{20}{2} = 10 \)
2. \( x = \frac{-10}{2} = -5 \)

Thus, the roots are \( a = 10 \) and \( b = -5 \).

### Step 5: Find \( a - b \)
Now we can find \( a - b \):
\[ a - b = 10 - (-5) = 10 + 5 = 15 \]

### Final Answer
The value of \( a - b \) is \( 15 \).

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