Which equation has the same solution as x^(2)+x+19=-2 ? (x+0.5)^(2)=-21.25 (x-0.5)^(2)=-21.25 (x-0.5)^(2)=-20.75 (x+0.5)^(2)=-20.75

Question

Which equation has the same solution as  x^(2)+x+19=-2 ?  (x+0.5)^(2)=-21.25  (x-0.5)^(2)=-21.25  (x-0.5)^(2)=-20.75  (x+0.5)^(2)=-20.75

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Simplify Equation: Simplify the given equation by moving all terms to one side to set the equation to zero. x^2 + x + 19 = -2 can be rewritten as x^2 + x + 21 = 0 by adding 2 to both sides.Identify a, b, c: Compare the given equation x^2 + x + 21 = 0 with the standard form of a quadratic equation ax^2 + bx + c = 0 to identify a, b, and c. Here, a = 1, b = 1, and c = 21.Compare with Choices: Look at the provided answer choices to determine which one can be rewritten in the form of x^2 + x + 21 = 0. We need to expand each choice and compare it with the given equation.Expand Choice 1: Expand the first choice (x + 0.5)^2 = -21.25. (x + 0.5)^2 = x^2 + 2*(0.5)*x + (0.5)^2 = x^2 + x + 0.25. This does not match the given equation x^2 + x + 21 = 0.Expand Choice 2: Expand the second choice (x - 0.5)^2 = -21.25. (x - 0.5)^2 = x^2 - 2*(0.5)*x + (0.5)^2 = x^2 - x + 0.25. This does not match the given equation x^2 + x + 21 = 0.Expand Choice 3: Expand the third choice (x - 0.5)^2 = -20.75. (x - 0.5)^2 = x^2 - 2*(0.5)*x + (0.5)^2 = x^2 - x + 0.25. This does not match the given equation x^2 + x + 21 = 0.Expand Choice 4: Expand the fourth choice (x + 0.5)^2 = -20.75. (x + 0.5)^2 = x^2 + 2*(0.5)*x + (0.5)^2 = x^2 + x + 0.25. Now, we need to move -20.75 to the other side to compare it with the given equation. x^2 + x + 0.25 = 20.75 x^2 + x + 21 = 0 (by adding 20.75 to both sides) This matches the given equation x^2 + x + 21 = 0.

Which equation has the same solution as $x^{2}+x+19=-2$ ? $\newline$ $(x+0.5)^{2}=-21.25$ $\newline$ $(x-0.5)^{2}=-21.25$ $\newline$ $(x-0.5)^{2}=-20.75$ $\newline$ $(x+0.5)^{2}=-20.75$

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