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Factor completely.

9x^(2)+6x+1=

Factor completely.\newline9x2+6x+1=9x^{2}+6x+1=

Full solution

Q. Factor completely.\newline9x2+6x+1=9x^{2}+6x+1=
  1. Determine Perfect Square Trinomial: Determine if the quadratic can be factored as a perfect square trinomial.\newlineA perfect square trinomial is in the form (ax)2+2abx+b2(ax)^2 + 2abx + b^2, which factors to (ax+b)2(ax + b)^2.\newlineWe can compare the given quadratic 9x2+6x+19x^2 + 6x + 1 with the perfect square trinomial form.
  2. Identify a2a^2, 2ab2ab, and b2b^2: Identify a2a^2, 2ab2ab, and b2b^2 in the expression 9x2+6x+19x^2 + 6x + 1.\newlinea2=9x2a^2 = 9x^2, which suggests a=3xa = 3x because (3x)2=9x2(3x)^2 = 9x^2.\newline2ab2ab00, which suggests 2ab2ab11 because 2ab2ab22.\newline2ab2ab should be 2ab2ab44, which matches the middle term of the quadratic.
  3. Factor using Perfect Square Trinomial Pattern: Factor the quadratic using the perfect square trinomial pattern.\newlineSince we have identified a=3xa = 3x and b=1b = 1, the factored form is $(ax + b)^\(2\) = (\(3\)x + \(1\))^\(2\).