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The equation of a parabola is y=x2+8x+19y = x^2 + 8x + 19. 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+8x+19y = x^2 + 8x + 19. 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 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 transform the given equation into vertex form.\newlineThe given equation is y=x2+8x+19y = x^2 + 8x + 19. To complete the square, we need to find the value that makes x2+8xx^2 + 8x into a perfect square trinomial. We do this by taking half of the coefficient of xx, which is 88, dividing it by 22 to get 44, and then squaring it to get 1616. We will add and subtract this value within the equation.
  3. Add and subtract: Add and subtract the value found in Step 22 to the equation.\newliney=x2+8x+19y = x^2 + 8x + 19 can be written as y=x2+8x+16+19−16y = x^2 + 8x + 16 + 19 - 16, by adding and subtracting 1616.
  4. Rewrite equation: Rewrite the equation grouping the perfect square trinomial and combining the constants. y=(x2+8x+16)+19−16y = (x^2 + 8x + 16) + 19 - 16 simplifies to y=(x+4)2+3y = (x + 4)^2 + 3, since x2+8x+16x^2 + 8x + 16 is a perfect square trinomial and 19−1619 - 16 equals 33.

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