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Simplifying 16x2 + 78x + 7 = 0 Reorder the terms: 7 + 78x + 16x2 = 0 Solving 7 + 78x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + 4.875x + x2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + 4.875x + -0.4375 + x2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + 4.875x + x2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + 4.875x + x2 = 0 + -0.4375 4.875x + x2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 4.875x + x2 = -0.4375 The x term is 4.875x. Take half its coefficient (2.4375). Square it (5.94140625) and add it to both sides. Add '5.94140625' to each side of the equation. 4.875x + 5.94140625 + x2 = -0.4375 + 5.94140625 Reorder the terms: 5.94140625 + 4.875x + x2 = -0.4375 + 5.94140625 Combine like terms: -0.4375 + 5.94140625 = 5.50390625 5.94140625 + 4.875x + x2 = 5.50390625 Factor a perfect square on the left side: (x + 2.4375)(x + 2.4375) = 5.50390625 Calculate the square root of the right side: 2.346040547 Break this problem into two subproblems by setting (x + 2.4375) equal to 2.346040547 and -2.346040547.Subproblem 1
x + 2.4375 = 2.346040547 Simplifying x + 2.4375 = 2.346040547 Reorder the terms: 2.4375 + x = 2.346040547 Solving 2.4375 + x = 2.346040547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = 2.346040547 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = 2.346040547 + -2.4375 x = 2.346040547 + -2.4375 Combine like terms: 2.346040547 + -2.4375 = -0.091459453 x = -0.091459453 Simplifying x = -0.091459453Subproblem 2
x + 2.4375 = -2.346040547 Simplifying x + 2.4375 = -2.346040547 Reorder the terms: 2.4375 + x = -2.346040547 Solving 2.4375 + x = -2.346040547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = -2.346040547 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = -2.346040547 + -2.4375 x = -2.346040547 + -2.4375 Combine like terms: -2.346040547 + -2.4375 = -4.783540547 x = -4.783540547 Simplifying x = -4.783540547Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.091459453, -4.783540547}
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