AI Solution

When we solve a quadratic like

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

the first step is often to factor out the greatest common factor (GCF)—the largest number that divides every coefficient.

Simplifies the numbers
Dividing every term by 2 changes the equation from
$2x^{2}-8x+6=0$
to
$x^{2}-4x+3=0.$
The coefficients are now smaller, so arithmetic (especially factoring) is easier and less error‑prone.
Preserves the solution set
Multiplying an equation by a non‑zero constant (or dividing by one) does not change its roots. If $x$ satisfies the original equation, it also satisfies the simplified one, and vice‑versa.
Makes factoring more obvious
With smaller numbers it’s easier to spot two integers whose product equals the constant term (here +3) and whose sum equals the linear coefficient (here –4). Those integers are –1 and –3, leading directly to
$(x-1)(x-3)=0.$
Avoids unnecessary work
If you tried to factor the original form directly, you’d have to find two numbers that multiply to $2 \times 6 = 12$ and add to $-8$ . That works, but the intermediate step (splitting the middle term) is more cumbersome than the clean factor‑out‑GCF approach.

You could still solve it by the quadratic formula or by “splitting the middle term,” but the algebra becomes lengthier:

Using the quadratic formula with $a=2,\;b=-8,\;c=6$ gives

=\frac{8\pm\sqrt{16}}{4} =\frac{8\pm4}{4},$$ which again yields $x=1$ or $x=3$. The result is the same, but the intermediate numbers are larger. --- **Bottom line:** Dividing by the GCF is a shortcut that makes the subsequent factoring step quicker and reduces the chance of arithmetic mistakes, while leaving the solutions unchanged. If you’d like a visual explanation of this process, the video below walks through why and how we factor out the GCF before solving a quadratic. ---

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