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The equation of a parabola is y=x2+8x+12y = x^2 + 8x + 12. 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+12y = x^2 + 8x + 12. 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 rewrite the given equation in vertex form.\newlineThe given equation is y=x2+8x+12y = x^2 + 8x + 12. To complete the square, we need to find the value that makes x2+8xx^2 + 8x into a perfect square trinomial. This value is (8/2)2=42=16(8/2)^2 = 4^2 = 16. We will add and subtract 1616 within the equation.
  3. Add and subtract: Add and subtract 1616 to the equation.y=x2+8x+12y = x^2 + 8x + 12 can be written as y=x2+8x+16−16+12y = x^2 + 8x + 16 - 16 + 12 by adding and subtracting 1616.
  4. Group terms, combine: Group the perfect square trinomial and combine the constants.\newlineNow, we group the terms to form a perfect square trinomial and combine the constants: y=(x2+8x+16)−16+12y = (x^2 + 8x + 16) - 16 + 12.
  5. Factor trinomial, simplify: Factor the perfect square trinomial and simplify the constant term.\newlineThe factored form of the perfect square trinomial is (x+4)2(x + 4)^2, and combining the constants −16+12-16 + 12 gives −4-4. So, the equation becomes y=(x+4)2−4y = (x + 4)^2 - 4.

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