We have the system of equations:

1. \( a + b^2 = 7 \)
2. \( 9b + a^2 = 27 \)

Let's solve this system step-by-step.

### Step 1: Solve for \( a \) in terms of \( b \) from the first equation
From equation (1):
\[
a = 7 - b^2
\]

### Step 2: Substitute \( a = 7 - b^2 \) into the second equation
Substitute this into equation (2):
\[
9b + (7 - b^2)^2 = 27
\]

Now expand the square on the left-hand side:
\[
9b + (49 - 14b^2 + b^4) = 27
\]
Simplify:
\[
9b + b^4 - 14b^2 + 49 = 27
\]
Move all terms to one side:
\[
b^4 - 14b^2 + 9b + 49 - 27 = 0
\]
\[
b^4 - 14b^2 + 9b + 22 = 0
\]

### Step 3: Solve for \( b \)
We now have a quartic equation:
\[
b^4 - 14b^2 + 9b + 22 = 0
\]

This equation is quite complex, so we would typically use numerical methods or a calculator to find the solutions. Let me solve for \( b \) numerically.

I'll calculate that.The solutions for \( b \) are:

\[
b_1 = -1, \quad b_2 = 2, \quad b_3 = -\frac{1}{2} + \frac{3\sqrt{5}}{2}, \quad b_4 = -\frac{1}{2} - \frac{3\sqrt{5}}{2}
\]

Now let's find the corresponding values of \( a \) for each solution of \( b \).

### Step 4: Find \( a \) for each \( b \)
Using \( a = 7 - b^2 \), we can calculate \( a \) for each value of \( b \).

I'll compute that now.The corresponding values of \( a \) for each \( b \) are:

1. For \( b = -1 \), \( a = 6 \)
2. For \( b = 2 \), \( a = 3 \)
3. For \( b = -\frac{1}{2} + \frac{3\sqrt{5}}{2} \), the expression is more complex:  
   \[
   a = 7 - \left(-\frac{1}{2} + \frac{3\sqrt{5}}{2}\right)^2
   \]
4. For \( b = -\frac{1}{2} - \frac{3\sqrt{5}}{2} \), the expression is:  
   \[
   a = 7 - \left(-\frac{1}{2} - \frac{3\sqrt{5}}{2}\right)^2
   \]

The exact form of the last two solutions can be left in terms of square roots or computed numerically if required.

Do you need further details on these expressions or numerical approximations?

Here are some related questions to explore further:
1. How would you solve this system of equations using matrices?
2. How can complex solutions be handled in similar systems?
3. What is the importance of checking all roots for validity in real-world problems?
4. How does this quartic equation arise in practical applications?
5. Can you graph the functions \( a + b^2 = 7 \) and \( 9b + a^2 = 27 \)?

**Tip**: When solving higher-degree polynomials, be cautious of extraneous roots introduced during factoring or squaring steps.

Explore the solution of the system of equations a + b^2 = 7 and 9b + a^2 = 27 using substitution and quartic equations. This detailed step-by-step guide covers algebraic techniques and numerical methods, suitable for Grades 10-12 students.

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