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Simplifying 65(x + -5) * x = 180 Reorder the terms: 65(-5 + x) * x = 180 Reorder the terms for easier multiplication: 65x(-5 + x) = 180 (-5 * 65x + x * 65x) = 180 (-325x + 65x2) = 180 Solving -325x + 65x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + -325x + 65x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + -325x + 65x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-36 + -65x + 13x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-36 + -65x + 13x2)' equal to zero and attempt to solve: Simplifying -36 + -65x + 13x2 = 0 Solving -36 + -65x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -2.769230769 + -5x + x2 = 0 Move the constant term to the right: Add '2.769230769' to each side of the equation. -2.769230769 + -5x + 2.769230769 + x2 = 0 + 2.769230769 Reorder the terms: -2.769230769 + 2.769230769 + -5x + x2 = 0 + 2.769230769 Combine like terms: -2.769230769 + 2.769230769 = 0.000000000 0.000000000 + -5x + x2 = 0 + 2.769230769 -5x + x2 = 0 + 2.769230769 Combine like terms: 0 + 2.769230769 = 2.769230769 -5x + x2 = 2.769230769 The x term is -5x. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x + 6.25 + x2 = 2.769230769 + 6.25 Reorder the terms: 6.25 + -5x + x2 = 2.769230769 + 6.25 Combine like terms: 2.769230769 + 6.25 = 9.019230769 6.25 + -5x + x2 = 9.019230769 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 9.019230769 Calculate the square root of the right side: 3.003203418 Break this problem into two subproblems by setting (x + -2.5) equal to 3.003203418 and -3.003203418.Subproblem 1
x + -2.5 = 3.003203418 Simplifying x + -2.5 = 3.003203418 Reorder the terms: -2.5 + x = 3.003203418 Solving -2.5 + x = 3.003203418 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 3.003203418 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 3.003203418 + 2.5 x = 3.003203418 + 2.5 Combine like terms: 3.003203418 + 2.5 = 5.503203418 x = 5.503203418 Simplifying x = 5.503203418Subproblem 2
x + -2.5 = -3.003203418 Simplifying x + -2.5 = -3.003203418 Reorder the terms: -2.5 + x = -3.003203418 Solving -2.5 + x = -3.003203418 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -3.003203418 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -3.003203418 + 2.5 x = -3.003203418 + 2.5 Combine like terms: -3.003203418 + 2.5 = -0.503203418 x = -0.503203418 Simplifying x = -0.503203418Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.503203418, -0.503203418}Solution
x = {5.503203418, -0.503203418}
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