If `(a + b) ^(2) = 100 and (a - b) = 4 ` then ab equal to:

Options

A. 116
B. 84
C. 21
D. 53

Answer

To solve the problem, we have the equations:

1. \((a + b)^2 = 100\)
2. \(a - b = 4\)

We need to find the value of \(ab\).

### Step 1: Simplify the first equation
From the first equation, we can take the square root of both sides:

\[ a + b = \sqrt{100} \]

Calculating the square root:

\[ a + b = 10 \]

### Step 2: Write down the equations
Now we have two equations:

1. \(a + b = 10\) (Equation 1)
2. \(a - b = 4\) (Equation 2)

### Step 3: Add the two equations
Next, we will add Equation 1 and Equation 2:

\[ (a + b) + (a - b) = 10 + 4 \]

This simplifies to:

\[ 2a = 14 \]

### Step 4: Solve for \(a\)
Now, divide both sides by 2 to find \(a\):

\[ a = \frac{14}{2} = 7 \]

### Step 5: Substitute \(a\) back to find \(b\)
Now that we have \(a\), we can substitute it back into Equation 1 to find \(b\):

\[ 7 + b = 10 \]

Subtract 7 from both sides:

\[ b = 10 - 7 = 3 \]

### Step 6: Calculate \(ab\)
Now we can find \(ab\):

\[ ab = 7 \times 3 = 21 \]

Thus, the value of \(ab\) is **21**.