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The equation of a parabola is y=x2+10x+27y = x^2 + 10x + 27. Write the equation in vertex form.\newlineWrite any numbers as integers or simplified proper or improper fractions.\newline______

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Q. The equation of a parabola is y=x2+10x+27y = x^2 + 10x + 27. Write the equation in vertex form.\newlineWrite any numbers as integers or simplified proper or improper fractions.\newline______
  1. Identify vertex form: Identify the vertex form of a parabola.\newlineThe vertex form of a parabola is given by the equation y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  2. Complete square: Complete the square to rewrite the given equation in vertex form.\newlineWe start with the given equation y=x2+10x+27y = x^2 + 10x + 27. To complete the square, we need to find a value that, when added and subtracted to the equation, forms a perfect square trinomial with x2+10xx^2 + 10x. This value is (10/2)2=25(10/2)^2 = 25. We add and subtract 2525 within the equation.\newliney=x2+10x+25+27−25y = x^2 + 10x + 25 + 27 - 25
  3. Factor and simplify: Factor the perfect square trinomial and simplify.\newlineNow we factor the trinomial x2+10x+25x^2 + 10x + 25 to get (x+5)2(x + 5)^2. We then combine the constants 27−2527 - 25 to get 22.\newliney=(x+5)2+2y = (x + 5)^2 + 2\newlineThis is the equation in vertex form, where the vertex is (−5,2)(-5, 2).

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