https://file.aiquickdraw.com/mathbot/file/1727045007004x7l6lc7u.jpg

The equation from the image is:

\[
2(x^2 + 2x - 3) = 4x(4x - 9)
\]

Let's solve this step by step.

### Step 1: Expand both sides
Expand both the left-hand side (LHS) and the right-hand side (RHS) of the equation.

#### LHS:
\[
2(x^2 + 2x - 3) = 2x^2 + 4x - 6
\]

#### RHS:
\[
4x(4x - 9) = 16x^2 - 36x
\]

So, the equation becomes:
\[
2x^2 + 4x - 6 = 16x^2 - 36x
\]

### Step 2: Bring all terms to one side
Move all terms to the left-hand side to set the equation equal to 0:

\[
2x^2 + 4x - 6 - 16x^2 + 36x = 0
\]

Combine like terms:

\[
(2x^2 - 16x^2) + (4x + 36x) - 6 = 0
\]
\[
-14x^2 + 40x - 6 = 0
\]

### Step 3: Solve the quadratic equation
The equation we now have is:
\[
-14x^2 + 40x - 6 = 0
\]

We can divide the entire equation by -2 to simplify:

\[
7x^2 - 20x + 3 = 0
\]

Now, we solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 7 \), \( b = -20 \), and \( c = 3 \).

### Step 4: Apply the quadratic formula
\[
x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(7)(3)}}{2(7)}
\]
\[
x = \frac{20 \pm \sqrt{400 - 84}}{14}
\]
\[
x = \frac{20 \pm \sqrt{316}}{14}
\]
\[
x = \frac{20 \pm \sqrt{316}}{14}
\]

Since \( \sqrt{316} \) is approximately 17.78, we get:

\[
x = \frac{20 \pm 17.78}{14}
\]

### Step 5: Calculate the two possible solutions
\[
x_1 = \frac{20 + 17.78}{14} \approx \frac{37.78}{14} \approx 2.7
\]
\[
x_2 = \frac{20 - 17.78}{14} \approx \frac{2.22}{14} \approx 0.16
\]

### Final solutions:
The solutions are approximately:
\[
x_1 \approx 2.7 \quad \text{and} \quad x_2 \approx 0.16
\]

Would you like any further details or clarification on this? 

Here are some related questions:
1. How do you factor a quadratic equation instead of using the quadratic formula?
2. What are some methods to check the solutions of a quadratic equation?
3. Can you solve the same quadratic equation graphically?
4. How can the discriminant of the quadratic equation be interpreted?
5. What are the steps to solve quadratic equations by completing the square?

**Tip:** Always simplify equations where possible to make solving easier. For example, dividing by a common factor early on can simplify complex equations.

Solve the equation 2(x^2 + 2x - 3) = 4x(4x - 9)

Learn how to solve the quadratic equation 2(x^2 + 2x - 3) = 4x(4x - 9) step by step. This detailed solution covers expanding, combining like terms, and applying the quadratic formula.

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