To solve the quadratic equation $x^2 - 4x + 2 = 0$ using the quadratic formula, we can apply the following formula:

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

Where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

For the equation $x^2 - 4x + 2 = 0$, we identify:

• $a = 1$
• $b = -4$
• $c = 2$

Now, substitute these values into the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(2)}}{2(1)}$

Simplify the terms:

$x = \frac{4 \pm \sqrt{16 - 8}}{2}$ $x = \frac{4 \pm \sqrt{8}}{2}$ $x = \frac{4 \pm 2\sqrt{2}}{2}$

Now, divide both terms in the numerator by 2:

$x = 2 \pm \sqrt{2}$

Thus, the solutions are:

$x = 2 + \sqrt{2} \quad \text{or} \quad x = 2 - \sqrt{2}$

These are the two solutions for $x$.

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Would you like further explanation on this? Let me know if you have any questions!

Here are some follow-up questions that expand on this topic:

1. How does the discriminant ($b^2 - 4ac$) affect the nature of the roots?
2. How can we graph the quadratic equation to visualize the solutions?
3. What happens if the discriminant is negative in a quadratic equation?
4. How can we solve a quadratic equation by factoring instead of using the quadratic formula?
5. What are the steps for solving a quadratic equation by completing the square?

Tip: The quadratic formula is a universal method for solving any quadratic equation, and it works even when the equation cannot be factored easily.