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Simplifying 5x(x + -2.5) + 15(x + -2.5) = 0 Reorder the terms: 5x(-2.5 + x) + 15(x + -2.5) = 0 (-2.5 * 5x + x * 5x) + 15(x + -2.5) = 0 (-12.5x + 5x2) + 15(x + -2.5) = 0 Reorder the terms: -12.5x + 5x2 + 15(-2.5 + x) = 0 -12.5x + 5x2 + (-2.5 * 15 + x * 15) = 0 -12.5x + 5x2 + (-37.5 + 15x) = 0 Reorder the terms: -37.5 + -12.5x + 15x + 5x2 = 0 Combine like terms: -12.5x + 15x = 2.5x -37.5 + 2.5x + 5x2 = 0 Solving -37.5 + 2.5x + 5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -7.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 0.5x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 0.5x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 0.5x + x2 = 0 + 7.5 0.5x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 0.5x + x2 = 7.5 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 7.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 7.5 + 0.0625 Combine like terms: 7.5 + 0.0625 = 7.5625 0.0625 + 0.5x + x2 = 7.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 7.5625 Calculate the square root of the right side: 2.75 Break this problem into two subproblems by setting (x + 0.25) equal to 2.75 and -2.75.Subproblem 1
x + 0.25 = 2.75 Simplifying x + 0.25 = 2.75 Reorder the terms: 0.25 + x = 2.75 Solving 0.25 + x = 2.75 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 2.75 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 2.75 + -0.25 x = 2.75 + -0.25 Combine like terms: 2.75 + -0.25 = 2.5 x = 2.5 Simplifying x = 2.5Subproblem 2
x + 0.25 = -2.75 Simplifying x + 0.25 = -2.75 Reorder the terms: 0.25 + x = -2.75 Solving 0.25 + x = -2.75 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -2.75 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -2.75 + -0.25 x = -2.75 + -0.25 Combine like terms: -2.75 + -0.25 = -3 x = -3 Simplifying x = -3Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.5, -3}
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