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Simplifying (2x + -7)(2x + -7) + -1(7x + -6)(5x + -6) = 0 Reorder the terms: (-7 + 2x)(2x + -7) + -1(7x + -6)(5x + -6) = 0 Reorder the terms: (-7 + 2x)(-7 + 2x) + -1(7x + -6)(5x + -6) = 0 Multiply (-7 + 2x) * (-7 + 2x) (-7(-7 + 2x) + 2x * (-7 + 2x)) + -1(7x + -6)(5x + -6) = 0 ((-7 * -7 + 2x * -7) + 2x * (-7 + 2x)) + -1(7x + -6)(5x + -6) = 0 ((49 + -14x) + 2x * (-7 + 2x)) + -1(7x + -6)(5x + -6) = 0 (49 + -14x + (-7 * 2x + 2x * 2x)) + -1(7x + -6)(5x + -6) = 0 (49 + -14x + (-14x + 4x2)) + -1(7x + -6)(5x + -6) = 0 Combine like terms: -14x + -14x = -28x (49 + -28x + 4x2) + -1(7x + -6)(5x + -6) = 0 Reorder the terms: 49 + -28x + 4x2 + -1(-6 + 7x)(5x + -6) = 0 Reorder the terms: 49 + -28x + 4x2 + -1(-6 + 7x)(-6 + 5x) = 0 Multiply (-6 + 7x) * (-6 + 5x) 49 + -28x + 4x2 + -1(-6(-6 + 5x) + 7x * (-6 + 5x)) = 0 49 + -28x + 4x2 + -1((-6 * -6 + 5x * -6) + 7x * (-6 + 5x)) = 0 49 + -28x + 4x2 + -1((36 + -30x) + 7x * (-6 + 5x)) = 0 49 + -28x + 4x2 + -1(36 + -30x + (-6 * 7x + 5x * 7x)) = 0 49 + -28x + 4x2 + -1(36 + -30x + (-42x + 35x2)) = 0 Combine like terms: -30x + -42x = -72x 49 + -28x + 4x2 + -1(36 + -72x + 35x2) = 0 49 + -28x + 4x2 + (36 * -1 + -72x * -1 + 35x2 * -1) = 0 49 + -28x + 4x2 + (-36 + 72x + -35x2) = 0 Reorder the terms: 49 + -36 + -28x + 72x + 4x2 + -35x2 = 0 Combine like terms: 49 + -36 = 13 13 + -28x + 72x + 4x2 + -35x2 = 0 Combine like terms: -28x + 72x = 44x 13 + 44x + 4x2 + -35x2 = 0 Combine like terms: 4x2 + -35x2 = -31x2 13 + 44x + -31x2 = 0 Solving 13 + 44x + -31x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -31 the coefficient of the squared term: Divide each side by '-31'. -0.4193548387 + -1.419354839x + x2 = 0 Move the constant term to the right: Add '0.4193548387' to each side of the equation. -0.4193548387 + -1.419354839x + 0.4193548387 + x2 = 0 + 0.4193548387 Reorder the terms: -0.4193548387 + 0.4193548387 + -1.419354839x + x2 = 0 + 0.4193548387 Combine like terms: -0.4193548387 + 0.4193548387 = 0.0000000000 0.0000000000 + -1.419354839x + x2 = 0 + 0.4193548387 -1.419354839x + x2 = 0 + 0.4193548387 Combine like terms: 0 + 0.4193548387 = 0.4193548387 -1.419354839x + x2 = 0.4193548387 The x term is -1.419354839x. Take half its coefficient (-0.7096774195). Square it (0.5036420397) and add it to both sides. Add '0.5036420397' to each side of the equation. -1.419354839x + 0.5036420397 + x2 = 0.4193548387 + 0.5036420397 Reorder the terms: 0.5036420397 + -1.419354839x + x2 = 0.4193548387 + 0.5036420397 Combine like terms: 0.4193548387 + 0.5036420397 = 0.9229968784 0.5036420397 + -1.419354839x + x2 = 0.9229968784 Factor a perfect square on the left side: (x + -0.7096774195)(x + -0.7096774195) = 0.9229968784 Calculate the square root of the right side: 0.960727265 Break this problem into two subproblems by setting (x + -0.7096774195) equal to 0.960727265 and -0.960727265.Subproblem 1
x + -0.7096774195 = 0.960727265 Simplifying x + -0.7096774195 = 0.960727265 Reorder the terms: -0.7096774195 + x = 0.960727265 Solving -0.7096774195 + x = 0.960727265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7096774195' to each side of the equation. -0.7096774195 + 0.7096774195 + x = 0.960727265 + 0.7096774195 Combine like terms: -0.7096774195 + 0.7096774195 = 0.0000000000 0.0000000000 + x = 0.960727265 + 0.7096774195 x = 0.960727265 + 0.7096774195 Combine like terms: 0.960727265 + 0.7096774195 = 1.6704046845 x = 1.6704046845 Simplifying x = 1.6704046845Subproblem 2
x + -0.7096774195 = -0.960727265 Simplifying x + -0.7096774195 = -0.960727265 Reorder the terms: -0.7096774195 + x = -0.960727265 Solving -0.7096774195 + x = -0.960727265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7096774195' to each side of the equation. -0.7096774195 + 0.7096774195 + x = -0.960727265 + 0.7096774195 Combine like terms: -0.7096774195 + 0.7096774195 = 0.0000000000 0.0000000000 + x = -0.960727265 + 0.7096774195 x = -0.960727265 + 0.7096774195 Combine like terms: -0.960727265 + 0.7096774195 = -0.2510498455 x = -0.2510498455 Simplifying x = -0.2510498455Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.6704046845, -0.2510498455}
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