If `a +b= 7 and ab= 10`, find `a-b`

Answer

To solve the problem where \( a + b = 7 \) and \( ab = 10 \), and we need to find \( a - b \), we can follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \( a + b = 7 \) (Equation 1)
2. \( ab = 10 \) (Equation 2)

### Step 2: Express \( a \) in terms of \( b \)
From Equation 1, we can express \( a \) as:
\[ a = 7 - b \]

### Step 3: Substitute \( a \) in Equation 2
Now, we substitute \( a \) in Equation 2:
\[ (7 - b)b = 10 \]

### Step 4: Expand the equation
Expanding the left side gives us:
\[ 7b - b^2 = 10 \]

### Step 5: Rearrange the equation
Rearranging the equation to bring all terms to one side gives:
\[ b^2 - 7b + 10 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor the quadratic equation \( b^2 - 7b + 10 = 0 \). We look for two numbers that multiply to \( 10 \) and add up to \( -7 \). The factors are \( -5 \) and \( -2 \):
\[ (b - 5)(b - 2) = 0 \]

### Step 7: Solve for \( b \)
Setting each factor to zero gives us:
1. \( b - 5 = 0 \) → \( b = 5 \)
2. \( b - 2 = 0 \) → \( b = 2 \)

### Step 8: Find corresponding values of \( a \)
Now we can find the corresponding values of \( a \) for each value of \( b \):
- If \( b = 5 \):
\[ a = 7 - 5 = 2 \]
- If \( b = 2 \):
\[ a = 7 - 2 = 5 \]

### Step 9: Calculate \( a - b \)
Now we can find \( a - b \) for both cases:
1. When \( a = 2 \) and \( b = 5 \):
\[ a - b = 2 - 5 = -3 \]
2. When \( a = 5 \) and \( b = 2 \):
\[ a - b = 5 - 2 = 3 \]

### Final Result
Thus, the values of \( a - b \) are \( -3 \) and \( 3 \).

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