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Simplifying x(x + -6.5) = 222 Reorder the terms: x(-6.5 + x) = 222 (-6.5 * x + x * x) = 222 (-6.5x + x2) = 222 Solving -6.5x + x2 = 222 Solving for variable 'x'. Reorder the terms: -222 + -6.5x + x2 = 222 + -222 Combine like terms: 222 + -222 = 0 -222 + -6.5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '222' to each side of the equation. -222 + -6.5x + 222 + x2 = 0 + 222 Reorder the terms: -222 + 222 + -6.5x + x2 = 0 + 222 Combine like terms: -222 + 222 = 0 0 + -6.5x + x2 = 0 + 222 -6.5x + x2 = 0 + 222 Combine like terms: 0 + 222 = 222 -6.5x + x2 = 222 The x term is -6.5x. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5x + 10.5625 + x2 = 222 + 10.5625 Reorder the terms: 10.5625 + -6.5x + x2 = 222 + 10.5625 Combine like terms: 222 + 10.5625 = 232.5625 10.5625 + -6.5x + x2 = 232.5625 Factor a perfect square on the left side: (x + -3.25)(x + -3.25) = 232.5625 Calculate the square root of the right side: 15.25 Break this problem into two subproblems by setting (x + -3.25) equal to 15.25 and -15.25.Subproblem 1
x + -3.25 = 15.25 Simplifying x + -3.25 = 15.25 Reorder the terms: -3.25 + x = 15.25 Solving -3.25 + x = 15.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = 15.25 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = 15.25 + 3.25 x = 15.25 + 3.25 Combine like terms: 15.25 + 3.25 = 18.5 x = 18.5 Simplifying x = 18.5Subproblem 2
x + -3.25 = -15.25 Simplifying x + -3.25 = -15.25 Reorder the terms: -3.25 + x = -15.25 Solving -3.25 + x = -15.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = -15.25 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = -15.25 + 3.25 x = -15.25 + 3.25 Combine like terms: -15.25 + 3.25 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.5, -12}
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