Question

1. $x^2 - 25 = 0$

$(1x - 5)(x + 5)$
example 3: factoring out a greatest commo

1. $2x^2 - 12x + 18 = 0$

$2x(1x - 6x + 9)$
$2x$

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(2x^2\), \(-12x\), and \(18\) is \(2\). So we factor out \(2\) from each term:
\(2x^2 - 12x + 18 = 2(x^2 - 6x + 9)\)

Step2: Factor the quadratic

The quadratic \(x^2 - 6x + 9\) is a perfect square trinomial, which factors as \((x - 3)^2\) (since \((a - b)^2 = a^2 - 2ab + b^2\), here \(a = x\), \(b = 3\), and \(-2ab=-6x\), \(b^2 = 9\)).
So putting it together, \(2x^2 - 12x + 18 = 2(x - 3)^2\)

Step3: Solve for \(x\)

Set \(2(x - 3)^2 = 0\). Divide both sides by \(2\): \((x - 3)^2 = 0\). Take the square root of both sides: \(x - 3 = 0\), so \(x = 3\) (with multiplicity 2).

Answer:

The factored form is \(2(x - 3)^2\) and the solution is \(x = 3\) (double root)