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Simplifying 84 = 1.5x2 + 1.5x Reorder the terms: 84 = 1.5x + 1.5x2 Solving 84 = 1.5x + 1.5x2 Solving for variable 'x'. Reorder the terms: 84 + -1.5x + -1.5x2 = 1.5x + -1.5x + 1.5x2 + -1.5x2 Combine like terms: 1.5x + -1.5x = 0.0 84 + -1.5x + -1.5x2 = 0.0 + 1.5x2 + -1.5x2 84 + -1.5x + -1.5x2 = 1.5x2 + -1.5x2 Combine like terms: 1.5x2 + -1.5x2 = 0.0 84 + -1.5x + -1.5x2 = 0.0 Begin completing the square. Divide all terms by -1.5 the coefficient of the squared term: Divide each side by '-1.5'. -56 + x + x2 = 0 Move the constant term to the right: Add '56' to each side of the equation. -56 + x + 56 + x2 = 0 + 56 Reorder the terms: -56 + 56 + x + x2 = 0 + 56 Combine like terms: -56 + 56 = 0 0 + x + x2 = 0 + 56 x + x2 = 0 + 56 Combine like terms: 0 + 56 = 56 x + x2 = 56 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 56 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 56 + 0.25 Combine like terms: 56 + 0.25 = 56.25 1.25 + x2 = 56.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (x + 0.5) equal to 7.5 and -7.5.Subproblem 1
x + 0.5 = 7.5 Simplifying x + 0.5 = 7.5 Reorder the terms: 0.5 + x = 7.5 Solving 0.5 + x = 7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 7.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.5 + -0.5 x = 7.5 + -0.5 Combine like terms: 7.5 + -0.5 = 7 x = 7 Simplifying x = 7Subproblem 2
x + 0.5 = -7.5 Simplifying x + 0.5 = -7.5 Reorder the terms: 0.5 + x = -7.5 Solving 0.5 + x = -7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -7.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.5 + -0.5 x = -7.5 + -0.5 Combine like terms: -7.5 + -0.5 = -8 x = -8 Simplifying x = -8Solution
The solution to the problem is based on the solutions from the subproblems. x = {7, -8}
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