Let's solve the equation step by step:

The given equation is:

\[
(4x)(4x) - 13x = 35
\]

### Step 1: Simplify the expression
First, simplify \((4x)(4x)\) to \(16x^2\):

\[
16x^2 - 13x = 35
\]

### Step 2: Rearrange the equation
Move all terms to one side to set the equation equal to 0:

\[
16x^2 - 13x - 35 = 0
\]

### Step 3: Solve the quadratic equation
We now have a quadratic equation:

\[
16x^2 - 13x - 35 = 0
\]

We can solve this using the **quadratic formula**:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 16\), \(b = -13\), and \(c = -35\).

Let's substitute the values into the formula:

\[
x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(16)(-35)}}{2(16)}
\]

\[
x = \frac{13 \pm \sqrt{169 + 2240}}{32}
\]

\[
x = \frac{13 \pm \sqrt{2409}}{32}
\]

\[
x = \frac{13 \pm 49.08}{32}
\]

### Step 4: Find the two possible values for \(x\)
Now, calculate both possible values for \(x\):

1. \(x_1 = \frac{13 + 49.08}{32} = \frac{62.08}{32} \approx 1.94\)
2. \(x_2 = \frac{13 - 49.08}{32} = \frac{-36.08}{32} \approx -1.13\)

### Solution:
The two solutions are:

\[
x \approx 1.94 \quad \text{or} \quad x \approx -1.13
\]

Would you like more details on any of the steps or have any questions?

Here are 5 related questions to explore:
1. How do you factor quadratic equations?
2. What is the discriminant, and how does it help in solving quadratics?
3. Can quadratic equations have complex solutions?
4. What are the possible types of solutions for a quadratic equation?
5. How can you graph quadratic equations?

**Tip:** Always check both solutions of a quadratic equation by plugging them back into the original equation.

Learn how to solve the quadratic equation (4x)(4x) - 13x = 35 with a detailed step-by-step solution. This problem explores core algebraic concepts like simplifying quadratic expressions and using the quadratic formula. Suitable for students in Grades 9-10.

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