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Answer

The solutions of

are

Step‑by‑step solution

1. Identify a common factor (GCF).
   All three terms are divisible by 2:

   $$2x^{2}-8x+6 = 2\bigl(x^{2}-4x+3\bigr).$$

   Since the factor 2 does not affect the roots (it would be zero only if the whole expression were zero, which it isn’t), we can divide the entire equation by 2 to simplify:

   $$x^{2}-4x+3 = 0 .$$

2. Factor the quadratic $x^{2}-4x+3$.
   We look for two numbers that multiply to the constant term 3 and add to the coefficient of the linear term $-4$.
   The pair $-1$ and $-3$ works because

   $$(-1)\times(-3)=3,\qquad (-1)+(-3)=-4 .$$

   Thus

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

3. Set each factor equal to zero (Zero‑Product Property).

   $$(x-1)=0 \quad \text{or} \quad (x-3)=0 .$$

   Solving each gives

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

4. Check the solutions (optional).
   Substitute $x=1$ and $x=3$ back into the original equation:

   • For $x=1$: $2(1)^{2}-8(1)+6 = 2-8+6 = 0$.
   • For $x=3$: $2(3)^{2}-8(3)+6 = 18-24+6 = 0$.

   Both satisfy the equation, confirming the solutions.

Why we divided by 2 first

Dividing every term by the greatest common factor (GCF) simplifies the coefficients, making factoring easier and reducing the chance of arithmetic errors. It’s equivalent to factoring the GCF out of the expression; the roots of the simplified quadratic are the same as those of the original.

Want a visual walkthrough?

Below is a short video that demonstrates the same steps with clear visuals.

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