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Answer

The solutions are

Step‑by‑step solution

1. Write the equation

   $$2x^{2} - 8x + 6 = 0$$

2. Factor out the common factor (the coefficient 2) to simplify:

   $$2\bigl(x^{2} - 4x + 3\bigr) = 0$$

   Since the factor 2 is never zero, we only need to solve the quadratic inside the parentheses.

3. Factor the quadratic $x^{2} - 4x + 3$.
   Look for two numbers that multiply to $+3$ and add to $-4$. Those numbers are $-1$ and $-3$:

   $$x^{2} - 4x + 3 = (x - 1)(x - 3)$$

4. Set each factor equal to zero (zero‑product property):

   $$\begin{cases} x - 1 = 0 \\ x - 3 = 0 \end{cases}$$

5. Solve for $x$:

   $$x = 1 \quad \text{or} \quad x = 3$$

Alternative: Quadratic formula

If you prefer the formula, start from the original equation $2x^{2} - 8x + 6 = 0$.

1. Identify $a = 2$, $b = -8$, $c = 6$.
2. Plug into $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$
3. Compute the discriminant: $$b^{2} - 4ac = (-8)^{2} - 4(2)(6) = 64 - 48 = 16$$
4. Continue: $$x = \frac{8 \pm \sqrt{16}}{4} = \frac{8 \pm 4}{4}$$
5. This gives the two solutions: $$x = \frac{8 + 4}{4} = 3 \quad\text{and}\quad x = \frac{8 - 4}{4} = 1$$

Both methods lead to the same answers.

Want a visual walkthrough? A short video that walks through these steps can reinforce the process.