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Simplifying 7.5x2 + -30x + 30 = x2 Reorder the terms: 30 + -30x + 7.5x2 = x2 Solving 30 + -30x + 7.5x2 = x2 Solving for variable 'x'. Combine like terms: 7.5x2 + -1x2 = 6.5x2 30 + -30x + 6.5x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 30 + -30x + 6.5x2 = 0 Begin completing the square. Divide all terms by 6.5 the coefficient of the squared term: Divide each side by '6.5'. 4.615384615 + -4.615384615x + x2 = 0 Move the constant term to the right: Add '-4.615384615' to each side of the equation. 4.615384615 + -4.615384615x + -4.615384615 + x2 = 0 + -4.615384615 Reorder the terms: 4.615384615 + -4.615384615 + -4.615384615x + x2 = 0 + -4.615384615 Combine like terms: 4.615384615 + -4.615384615 = 0.000000000 0.000000000 + -4.615384615x + x2 = 0 + -4.615384615 -4.615384615x + x2 = 0 + -4.615384615 Combine like terms: 0 + -4.615384615 = -4.615384615 -4.615384615x + x2 = -4.615384615 The x term is -4.615384615x. Take half its coefficient (-2.307692308). Square it (5.325443788) and add it to both sides. Add '5.325443788' to each side of the equation. -4.615384615x + 5.325443788 + x2 = -4.615384615 + 5.325443788 Reorder the terms: 5.325443788 + -4.615384615x + x2 = -4.615384615 + 5.325443788 Combine like terms: -4.615384615 + 5.325443788 = 0.710059173 5.325443788 + -4.615384615x + x2 = 0.710059173 Factor a perfect square on the left side: (x + -2.307692308)(x + -2.307692308) = 0.710059173 Calculate the square root of the right side: 0.842650089 Break this problem into two subproblems by setting (x + -2.307692308) equal to 0.842650089 and -0.842650089.Subproblem 1
x + -2.307692308 = 0.842650089 Simplifying x + -2.307692308 = 0.842650089 Reorder the terms: -2.307692308 + x = 0.842650089 Solving -2.307692308 + x = 0.842650089 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + x = 0.842650089 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + x = 0.842650089 + 2.307692308 x = 0.842650089 + 2.307692308 Combine like terms: 0.842650089 + 2.307692308 = 3.150342397 x = 3.150342397 Simplifying x = 3.150342397Subproblem 2
x + -2.307692308 = -0.842650089 Simplifying x + -2.307692308 = -0.842650089 Reorder the terms: -2.307692308 + x = -0.842650089 Solving -2.307692308 + x = -0.842650089 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + x = -0.842650089 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + x = -0.842650089 + 2.307692308 x = -0.842650089 + 2.307692308 Combine like terms: -0.842650089 + 2.307692308 = 1.465042219 x = 1.465042219 Simplifying x = 1.465042219Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.150342397, 1.465042219}
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