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Let's solve the equation using the quadratic formula step by step. The given equation is:

\[
2x^2 - 4x = 1
\]

We first rearrange it to standard quadratic form \( ax^2 + bx + c = 0 \):

\[
2x^2 - 4x - 1 = 0
\]

Now, identify the coefficients:
- \( a = 2 \)
- \( b = -4 \)
- \( c = -1 \)

The quadratic formula is:

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

### Step 1: Calculate the discriminant
\[
\Delta = b^2 - 4ac = (-4)^2 - 4(2)(-1) = 16 + 8 = 24
\]

### Step 2: Apply the quadratic formula
\[
x = \frac{-(-4) \pm \sqrt{24}}{2(2)} = \frac{4 \pm \sqrt{24}}{4}
\]

### Step 3: Simplify the square root
\[
\sqrt{24} \approx 4.899
\]

Thus, the solutions are:

\[
x = \frac{4 + 4.899}{4} = \frac{8.899}{4} \approx 2.22
\]
\[
x = \frac{4 - 4.899}{4} = \frac{-0.899}{4} \approx -0.22
\]

### Final Answer:
The two solutions, rounded to the nearest hundredth, are:

\[
x \approx 2.22, -0.22
\]

Would you like any further explanation or details on this solution?

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Here are 5 related questions to explore further:
1. How does the discriminant help in determining the number of solutions?
2. What happens when the discriminant is zero?
3. Can you solve other quadratic equations with different methods like factoring or completing the square?
4. How do different values of \( a \), \( b \), and \( c \) affect the shape of the parabola?
5. Can the quadratic formula be derived from completing the square?

**Tip:** Always check the discriminant first to understand the nature of the solutions (real or complex).

Use the quadratic formula to solve for x: 2x² - 4x = 1. Round your answer to the nearest hundredth. If there is more than one solution, separate them with commas.

Learn how to solve the quadratic equation 2x² - 4x = 1 step-by-step using the quadratic formula. This problem involves algebra, discriminants, and square roots, and is suitable for students in Grades 8-10. Find the solutions rounded to the nearest hundredth.

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