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To solve the equation by factoring and applying the zero-product property, let’s go through the steps:

Given equation:
\[
(x^2 - 6x + 3)(2x^2 - 4x - 7) = 0
\]

This is a product of two expressions set equal to zero, so we can apply the **Zero-Product Property** here, which states that if \( A \cdot B = 0 \), then either \( A = 0 \) or \( B = 0 \).

### Step 1: Apply the Zero-Product Property
Set each factor equal to zero:
1. \( x^2 - 6x + 3 = 0 \)
2. \( 2x^2 - 4x - 7 = 0 \)

### Step 2: Solve Each Equation Individually

#### Solving \( x^2 - 6x + 3 = 0 \)
Since this is a quadratic equation, we can use the **Quadratic Formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For \( x^2 - 6x + 3 = 0 \), \( a = 1 \), \( b = -6 \), and \( c = 3 \).

Plugging these values into the formula:
\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}
\]
\[
x = \frac{6 \pm \sqrt{36 - 12}}{2}
\]
\[
x = \frac{6 \pm \sqrt{24}}{2}
\]
\[
x = \frac{6 \pm 2\sqrt{6}}{2}
\]
\[
x = 3 \pm \sqrt{6}
\]
So, the solutions for \( x^2 - 6x + 3 = 0 \) are:
\[
x = 3 + \sqrt{6} \quad \text{and} \quad x = 3 - \sqrt{6}
\]

#### Solving \( 2x^2 - 4x - 7 = 0 \)
Again, use the Quadratic Formula. For \( 2x^2 - 4x - 7 = 0 \), \( a = 2 \), \( b = -4 \), and \( c = -7 \).

Plugging these values into the formula:
\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-7)}}{2 \cdot 2}
\]
\[
x = \frac{4 \pm \sqrt{16 + 56}}{4}
\]
\[
x = \frac{4 \pm \sqrt{72}}{4}
\]
\[
x = \frac{4 \pm 6\sqrt{2}}{4}
\]
\[
x = 1 \pm \frac{3\sqrt{2}}{2}
\]
So, the solutions for \( 2x^2 - 4x - 7 = 0 \) are:
\[
x = 1 + \frac{3\sqrt{2}}{2} \quad \text{and} \quad x = 1 - \frac{3\sqrt{2}}{2}
\]

### Step 3: List the Solutions from Least to Greatest
The solutions, in ascending order, are:
\[
x = 1 - \frac{3\sqrt{2}}{2}, \quad x = 3 - \sqrt{6}, \quad x = 3 + \sqrt{6}, \quad x = 1 + \frac{3\sqrt{2}}{2}
\]

Would you like further clarification on any of these steps?

---

### Related Questions:
1. How do you derive the quadratic formula from a general quadratic equation?
2. What are other methods to solve quadratic equations besides factoring and using the quadratic formula?
3. How does the zero-product property work in polynomials with more than two factors?
4. What is the process to simplify square roots when they appear in solutions?
5. How can we approximate the square root values to list solutions numerically?

### Tip:
When solving quadratics, always check if factoring is possible first—it can simplify the problem compared to using the quadratic formula.

Solve the equation by factoring and/or the quadratic formula. Write "none" when needed. Copy and paste ±√ as needed. (x^2 - 6x + 3)(2x^2 - 4x - 7) = 0

Step-by-step solution to solve the quadratic equation (x^2 - 6x + 3)(2x^2 - 4x - 7) = 0 by factoring and the quadratic formula. This problem covers key algebraic techniques, including the zero-product property and quadratic formula application. Suitable for high school students in Grades 9-12.

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