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Simplifying 0 = 5x2 + 3.25x + -5.25 Reorder the terms: 0 = -5.25 + 3.25x + 5x2 Solving 0 = -5.25 + 3.25x + 5x2 Solving for variable 'x'. Combine like terms: 0 + 5.25 = 5.25 5.25 + -3.25x + -5x2 = -5.25 + 3.25x + 5x2 + 5.25 + -3.25x + -5x2 Reorder the terms: 5.25 + -3.25x + -5x2 = -5.25 + 5.25 + 3.25x + -3.25x + 5x2 + -5x2 Combine like terms: -5.25 + 5.25 = 0.00 5.25 + -3.25x + -5x2 = 0.00 + 3.25x + -3.25x + 5x2 + -5x2 5.25 + -3.25x + -5x2 = 3.25x + -3.25x + 5x2 + -5x2 Combine like terms: 3.25x + -3.25x = 0.00 5.25 + -3.25x + -5x2 = 0.00 + 5x2 + -5x2 5.25 + -3.25x + -5x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 5.25 + -3.25x + -5x2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1.05 + 0.65x + x2 = 0 Move the constant term to the right: Add '1.05' to each side of the equation. -1.05 + 0.65x + 1.05 + x2 = 0 + 1.05 Reorder the terms: -1.05 + 1.05 + 0.65x + x2 = 0 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + 0.65x + x2 = 0 + 1.05 0.65x + x2 = 0 + 1.05 Combine like terms: 0 + 1.05 = 1.05 0.65x + x2 = 1.05 The x term is 0.65x. Take half its coefficient (0.325). Square it (0.105625) and add it to both sides. Add '0.105625' to each side of the equation. 0.65x + 0.105625 + x2 = 1.05 + 0.105625 Reorder the terms: 0.105625 + 0.65x + x2 = 1.05 + 0.105625 Combine like terms: 1.05 + 0.105625 = 1.155625 0.105625 + 0.65x + x2 = 1.155625 Factor a perfect square on the left side: (x + 0.325)(x + 0.325) = 1.155625 Calculate the square root of the right side: 1.075 Break this problem into two subproblems by setting (x + 0.325) equal to 1.075 and -1.075.Subproblem 1
x + 0.325 = 1.075 Simplifying x + 0.325 = 1.075 Reorder the terms: 0.325 + x = 1.075 Solving 0.325 + x = 1.075 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.325' to each side of the equation. 0.325 + -0.325 + x = 1.075 + -0.325 Combine like terms: 0.325 + -0.325 = 0.000 0.000 + x = 1.075 + -0.325 x = 1.075 + -0.325 Combine like terms: 1.075 + -0.325 = 0.75 x = 0.75 Simplifying x = 0.75Subproblem 2
x + 0.325 = -1.075 Simplifying x + 0.325 = -1.075 Reorder the terms: 0.325 + x = -1.075 Solving 0.325 + x = -1.075 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.325' to each side of the equation. 0.325 + -0.325 + x = -1.075 + -0.325 Combine like terms: 0.325 + -0.325 = 0.000 0.000 + x = -1.075 + -0.325 x = -1.075 + -0.325 Combine like terms: -1.075 + -0.325 = -1.4 x = -1.4 Simplifying x = -1.4Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.75, -1.4}
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