QuestionFactor the following trinomial completely. $4 a^{2}-4 a+1$

Studdy Solution

STEP 1

1. The expression is a quadratic trinomial.
2. We are asked to factor the trinomial completely.
3. The trinomial may be a perfect square trinomial.

STEP 2

1. Identify if the trinomial is a perfect square.
2. Use the perfect square trinomial formula to factor the expression.

STEP 3

Check if the trinomial $4a^2 - 4a + 1$ is a perfect square trinomial. A perfect square trinomial has the form $(A - B)^2 = A^2 - 2AB + B^2$ .
Identify $A$ and $B$ such that: - $A^2 = 4a^2$ - $B^2 = 1$ - $-2AB = -4a$
From $A^2 = 4a^2$ , we have $A = 2a$ .
From $B^2 = 1$ , we have $B = 1$ .
Check if $-2AB = -4a$ holds: $-2(2a)(1) = -4a$ This is true, so the trinomial is a perfect square.

STEP 4

Use the perfect square trinomial formula to factor the expression:
Since $4a^2 - 4a + 1$ is a perfect square trinomial, it can be factored as: $(2a - 1)^2$

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