Question

factoring
$4x^2 - 20x + 25 = 0$

Explanation:

Step1: Identify the quadratic form

The equation is \(4x^2 - 20x + 25 = 0\) (assuming the middle term is \(-20x\) as the original might have a typo, since \(4x^2 - 20x + 25\) is a perfect square trinomial). A perfect square trinomial has the form \(a^2 - 2ab + b^2=(a - b)^2\). Here, \(a = 2x\) (since \((2x)^2 = 4x^2\)) and \(b = 5\) (since \(b^2=25\) and \(2ab = 2\times(2x)\times5 = 20x\)).

Step2: Factor the quadratic

Using the perfect square trinomial formula, \(4x^2 - 20x + 25=(2x - 5)^2\). So the factored form of \(4x^2 - 20x + 25 = 0\) is \((2x - 5)^2=0\).

Step3: Solve for x

If \((2x - 5)^2 = 0\), then \(2x - 5 = 0\) (since a square of a number is zero only if the number itself is zero). Solving \(2x - 5 = 0\) gives \(2x=5\), so \(x=\frac{5}{2}\).

Answer:

The factored form is \((2x - 5)^2 = 0\) and the solution is \(x=\frac{5}{2}\) (with multiplicity 2).