Vertex of y = x^2 + 2x — step-by-step solution (vertex = (-1, -1))

Find the vertex of the parabola y = x^2 + 2x. Step-by-step methods using the vertex formula and completing the square. Simplified coordinates: (-1, -1).

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We want the vertex of y = x^2 + 2x.

  1. Method 1 — Vertex formula: For a quadratic y = ax^2 + bx + c, the x-coordinate of the vertex is x = -b/(2a). Here a = 1 and b = 2, so

x_v = -2/(2·1) = -1.

Now evaluate y at x = -1:

y_v = (-1)^2 + 2(-1) = 1 - 2 = -1.

  1. Method 2 — Completing the square: Rewrite the quadratic:

x^2 + 2x = (x^2 + 2x + 1) - 1 = (x + 1)^2 - 1.

This is in the form (x - h)^2 + k with h = -1 and k = -1, so the vertex is (h, k) = (-1, -1).

Both coordinates are integers. Final answer: (-1, -1).

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