9x^2+24x+c In the given expression, c is constant. If the expression can be factored to the form (ax+b)^2, what is the value of c

We are given the quadratic expression \( 9x^2 + 24x + c \), and we are told that it can be factored into the form \( (ax + b)^2 \). 

### Step 1: Expand the factored form
The expanded form of \( (ax + b)^2 \) is:
\[
(ax + b)^2 = a^2x^2 + 2abx + b^2
\]
We want to match this expanded form with the original expression \( 9x^2 + 24x + c \).

### Step 2: Compare coefficients
From the expanded form, we see that:
- The coefficient of \( x^2 \) is \( a^2 \),
- The coefficient of \( x \) is \( 2ab \),
- The constant term is \( b^2 \).

Now, comparing the coefficients of \( 9x^2 + 24x + c \) with those from the expanded form:

1. The coefficient of \( x^2 \) gives: 
   \[
   a^2 = 9 \quad \Rightarrow \quad a = 3 \, (\text{since} \, a > 0)
   \]
   
2. The coefficient of \( x \) gives:
   \[
   2ab = 24 \quad \Rightarrow \quad 2(3)b = 24 \quad \Rightarrow \quad 6b = 24 \quad \Rightarrow \quad b = 4
   \]

### Step 3: Solve for \( c \)
The constant term from the expanded form is \( b^2 \). Therefore:
\[
c = b^2 = 4^2 = 16
\]

Thus, the value of \( c \) is \( \boxed{16} \).

Would you like to see more details or clarifications?

Here are some follow-up questions you might find helpful:
1. What is the process of factoring a general quadratic expression?
2. How do we handle cases where the quadratic cannot be factored into a perfect square?
3. What are other methods for solving quadratics, such as completing the square or using the quadratic formula?
4. How would the solution change if the quadratic were not a perfect square trinomial?
5. Can every quadratic expression be factored in real numbers?

**Tip:** Always check if a quadratic can be factored into a perfect square by looking at its discriminant \( \Delta = b^2 - 4ac \). If \( \Delta = 0 \), it can be factored as a perfect square.

9x^2 + 24x + c. In the given expression, c is constant. If the expression can be factored to the form (ax + b)^2, what is the value of c?

Discover how to factor the quadratic expression 9x^2 + 24x + c into a perfect square trinomial. This step-by-step guide explores the algebraic methods used to find the value of c and teaches key concepts such as quadratic equations and factoring techniques. Ideal for students in Grades 9-10.

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