https://file.aiquickdraw.com/mathbot/file/1728512482123t9egtpgc.jpg

The equation shown in the image is:

\[
2^{3x+2} - 3(2^{2x+3}) - 6(2^x) - 24 = 0
\]

Let's break this down step by step.

### Step 1: Substitute \( y = 2^x \)

This substitution helps simplify the exponential terms. The equation becomes:
\[
2^{3x+2} = 2^2 \cdot (2^x)^3 = 4y^3
\]
\[
2^{2x+3} = 2^3 \cdot (2^x)^2 = 8y^2
\]
Now the equation is:
\[
4y^3 - 3(8y^2) - 6y - 24 = 0
\]
Simplify:
\[
4y^3 - 24y^2 - 6y - 24 = 0
\]

### Step 2: Factor the cubic equation

We will attempt to factor this cubic equation. First, we check for any obvious roots by trial and error, plugging small integer values for \( y \).

Testing \( y = 2 \):
\[
4(2)^3 - 24(2)^2 - 6(2) - 24 = 4(8) - 24(4) - 12 - 24 = 32 - 96 - 12 - 24 = -100 \, (\text{no root})
\]

Testing \( y = 3 \):
\[
4(1)^3 - 24(1)^2 - 6(1)- button formLet's proceed carefully to solve this.

The equation provided in the image is:

\[
2^{3x+2} - 3(2^{2x+3}) - 6(2^x) - 24 = 0
\]

### Step 1: Substitution

Let \( y = 2^x \). This will simplify the equation:

- \( 2^{3x+2} = 2^2 \cdot (2^x)^3 = 4y^3 \)
- \( 2^{2x+3} = 2^3 \cdot (2^x)^2 = 8y^2 \)

Substitute these into the equation:

\[
4y^3 - 3(8y^2) - 6y - 24 = 0
\]

Simplify:

\[
4y^3 - 24y^2 - 6y - 24 = 0
\]

### Step 2: Factor the cubic equation

We now try to factor the cubic equation \( 4y^3 - 24y^2 - 6y - 24 = 0 \).

We start by factoring out a common factor of 2:

\[
2(2y^3 - 12y^2 - 3y - 12) = 0
\]

This simplifies to:

\[
2y^3 - 12y^2 - 3y - 12 = 0
\]

We attempt to factor this cubic equation using trial and error or synthetic division. Testing for small integer roots like \( y = 1 \), \( y = -1 \), etc., we find that \( y = -1 \) is a root.

### Step 3: Factor by synthetic division

Now that \( y = -1 \) is a root, we divide \( 2y^3 - 12y^2 - 3y - 12 \) by \( y + 1 \). After synthetic division, we get:

\[
2y^2 - 14y - 12 = 0
\]

### Step 4: Solve the quadratic equation

Solve the quadratic equation \( 2y^2 - 14y - 12 = 0 \) using the quadratic formula:

\[
y = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(2)(-12)}}{2(2)}
\]
\[
y = \frac{14 \pm \sqrt{196 + 96}}{4}
\]
\[
y = \frac{14 \pm \sqrt{292}}{4}
\]
\[
y = \frac{14 \pm 17.09}{4}
\]

Thus, the two roots are:

\[
y_1 = \frac{14 + 17.09}{4} \approx 7.77
\]
\[
y_2 = \frac{14 - 17.09}{4} \approx -0.77
\]

### Step 5: Solve for \( x \)

Now, recall that \( y = 2^x \).

For \( y = 7.77 \):

\[
2^x = 7.77 \quad \Rightarrow \quad x = \log_2(7.77) \approx 2.95
\]

For \( y = -0.77 \), there is no real solution since \( 2^x \) cannot be negative.

### Final Solution:

The only real solution is:

\[
x \approx 2.95
\]

Would you like further details or have any questions? Here are 5 related questions:

1. How does substitution help simplify exponential equations?
2. Can every cubic equation be factored easily?
3. What is synthetic division and how is it used in solving polynomial equations?
4. Why can't \( 2^x \) be negative in this context?
5. How can we use logarithms to solve for \( x \) in exponential equations?

**Tip**: When solving exponential equations, using substitution like \( y = 2^x \) or \( y = e^x \) is often helpful to reduce the complexity.

Solve the equation: 2^{3x+2} - 3(2^{2x+3}) - 6(2^x) - 24 = 0

Learn how to solve the exponential equation 2^{3x+2} - 3(2^{2x+3}) - 6(2^x) - 24 = 0 using algebraic substitution and factoring techniques. This problem involves converting exponential terms to a cubic equation, simplifying it, and solving for x. Suitable for Grades 10-12 students.

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