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Simplifying 4(3x2 + -5x) + 6 = 5(7x2 + -8) Reorder the terms: 4(-5x + 3x2) + 6 = 5(7x2 + -8) (-5x * 4 + 3x2 * 4) + 6 = 5(7x2 + -8) (-20x + 12x2) + 6 = 5(7x2 + -8) Reorder the terms: 6 + -20x + 12x2 = 5(7x2 + -8) Reorder the terms: 6 + -20x + 12x2 = 5(-8 + 7x2) 6 + -20x + 12x2 = (-8 * 5 + 7x2 * 5) 6 + -20x + 12x2 = (-40 + 35x2) Solving 6 + -20x + 12x2 = -40 + 35x2 Solving for variable 'x'. Reorder the terms: 6 + 40 + -20x + 12x2 + -35x2 = -40 + 35x2 + 40 + -35x2 Combine like terms: 6 + 40 = 46 46 + -20x + 12x2 + -35x2 = -40 + 35x2 + 40 + -35x2 Combine like terms: 12x2 + -35x2 = -23x2 46 + -20x + -23x2 = -40 + 35x2 + 40 + -35x2 Reorder the terms: 46 + -20x + -23x2 = -40 + 40 + 35x2 + -35x2 Combine like terms: -40 + 40 = 0 46 + -20x + -23x2 = 0 + 35x2 + -35x2 46 + -20x + -23x2 = 35x2 + -35x2 Combine like terms: 35x2 + -35x2 = 0 46 + -20x + -23x2 = 0 Begin completing the square. Divide all terms by -23 the coefficient of the squared term: Divide each side by '-23'. -2 + 0.8695652174x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.8695652174x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + 0.8695652174x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.8695652174x + x2 = 0 + 2 0.8695652174x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.8695652174x + x2 = 2 The x term is 0.8695652174x. Take half its coefficient (0.4347826087). Square it (0.1890359168) and add it to both sides. Add '0.1890359168' to each side of the equation. 0.8695652174x + 0.1890359168 + x2 = 2 + 0.1890359168 Reorder the terms: 0.1890359168 + 0.8695652174x + x2 = 2 + 0.1890359168 Combine like terms: 2 + 0.1890359168 = 2.1890359168 0.1890359168 + 0.8695652174x + x2 = 2.1890359168 Factor a perfect square on the left side: (x + 0.4347826087)(x + 0.4347826087) = 2.1890359168 Calculate the square root of the right side: 1.479539089 Break this problem into two subproblems by setting (x + 0.4347826087) equal to 1.479539089 and -1.479539089.Subproblem 1
x + 0.4347826087 = 1.479539089 Simplifying x + 0.4347826087 = 1.479539089 Reorder the terms: 0.4347826087 + x = 1.479539089 Solving 0.4347826087 + x = 1.479539089 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4347826087' to each side of the equation. 0.4347826087 + -0.4347826087 + x = 1.479539089 + -0.4347826087 Combine like terms: 0.4347826087 + -0.4347826087 = 0.0000000000 0.0000000000 + x = 1.479539089 + -0.4347826087 x = 1.479539089 + -0.4347826087 Combine like terms: 1.479539089 + -0.4347826087 = 1.0447564803 x = 1.0447564803 Simplifying x = 1.0447564803Subproblem 2
x + 0.4347826087 = -1.479539089 Simplifying x + 0.4347826087 = -1.479539089 Reorder the terms: 0.4347826087 + x = -1.479539089 Solving 0.4347826087 + x = -1.479539089 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4347826087' to each side of the equation. 0.4347826087 + -0.4347826087 + x = -1.479539089 + -0.4347826087 Combine like terms: 0.4347826087 + -0.4347826087 = 0.0000000000 0.0000000000 + x = -1.479539089 + -0.4347826087 x = -1.479539089 + -0.4347826087 Combine like terms: -1.479539089 + -0.4347826087 = -1.9143216977 x = -1.9143216977 Simplifying x = -1.9143216977Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.0447564803, -1.9143216977}
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