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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 the 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 the value that makes x2−10xx^2 - 10x a perfect square trinomial. This value is given by (b/2)2(b/2)^2, where bb is the coefficient of xx. In this case, b=−10b = -10, so (b/2)2=(−10/2)2=(−5)2=25(b/2)^2 = (-10/2)^2 = (-5)^2 = 25. We will add and subtract this value inside the equation.
  3. Add and Subtract Value: Add and subtract the value found in Step 22 to the equation.\newliney=x2−10x+25+27−25y = x^2 - 10x + 25 + 27 - 25\newlineThis allows us to write the equation as a perfect square trinomial plus a constant.
  4. Factor and Simplify: Factor the perfect square trinomial and simplify the constants.\newliney=(x2−10x+25)+27−25y = (x^2 - 10x + 25) + 27 - 25\newliney=(x−5)2+2y = (x - 5)^2 + 2\newlineNow the equation is in vertex form, where (h,k)=(5,2)(h, k) = (5, 2).

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