If one root of the equation `x^(2)+ax+8=0` is 4 while the equation `x^(2)+ax+b=0` has equal roots, find b.

Answer

To solve the problem step by step, we need to find the value of \( b \) given the conditions of the quadratic equations.

### Step 1: Use the first equation to find \( a \)

We know that one root of the equation \( x^2 + ax + 8 = 0 \) is 4. We can substitute \( x = 4 \) into the equation:

\[ 4^2 + a(4) + 8 = 0 \]

Calculating this gives:

\[ 16 + 4a + 8 = 0 \]

Combining like terms:

\[ 24 + 4a = 0 \]

Now, we can isolate \( a \):

\[ 4a = -24 \]
\[ a = -6 \]

### Step 2: Substitute \( a \) into the second equation

Now that we have \( a = -6 \), we can substitute this value into the second equation \( x^2 + ax + b = 0 \):

\[ x^2 - 6x + b = 0 \]

### Step 3: Use the condition of equal roots

For the equation to have equal roots, the discriminant must be zero. The discriminant \( D \) for a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:

\[ D = B^2 - 4AC \]

In our case, \( A = 1 \), \( B = -6 \), and \( C = b \). Therefore, we have:

\[ D = (-6)^2 - 4(1)(b) = 0 \]

Calculating this gives:

\[ 36 - 4b = 0 \]

### Step 4: Solve for \( b \)

Now we can solve for \( b \):

\[ 36 = 4b \]
\[ b = \frac{36}{4} = 9 \]

### Conclusion

The value of \( b \) is \( 9 \).

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