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Simplifying 5(2x + 3) = 3x(4x + 1) + -2(3x + 2) Reorder the terms: 5(3 + 2x) = 3x(4x + 1) + -2(3x + 2) (3 * 5 + 2x * 5) = 3x(4x + 1) + -2(3x + 2) (15 + 10x) = 3x(4x + 1) + -2(3x + 2) Reorder the terms: 15 + 10x = 3x(1 + 4x) + -2(3x + 2) 15 + 10x = (1 * 3x + 4x * 3x) + -2(3x + 2) 15 + 10x = (3x + 12x2) + -2(3x + 2) Reorder the terms: 15 + 10x = 3x + 12x2 + -2(2 + 3x) 15 + 10x = 3x + 12x2 + (2 * -2 + 3x * -2) 15 + 10x = 3x + 12x2 + (-4 + -6x) Reorder the terms: 15 + 10x = -4 + 3x + -6x + 12x2 Combine like terms: 3x + -6x = -3x 15 + 10x = -4 + -3x + 12x2 Solving 15 + 10x = -4 + -3x + 12x2 Solving for variable 'x'. Reorder the terms: 15 + 4 + 10x + 3x + -12x2 = -4 + -3x + 12x2 + 4 + 3x + -12x2 Combine like terms: 15 + 4 = 19 19 + 10x + 3x + -12x2 = -4 + -3x + 12x2 + 4 + 3x + -12x2 Combine like terms: 10x + 3x = 13x 19 + 13x + -12x2 = -4 + -3x + 12x2 + 4 + 3x + -12x2 Reorder the terms: 19 + 13x + -12x2 = -4 + 4 + -3x + 3x + 12x2 + -12x2 Combine like terms: -4 + 4 = 0 19 + 13x + -12x2 = 0 + -3x + 3x + 12x2 + -12x2 19 + 13x + -12x2 = -3x + 3x + 12x2 + -12x2 Combine like terms: -3x + 3x = 0 19 + 13x + -12x2 = 0 + 12x2 + -12x2 19 + 13x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 19 + 13x + -12x2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -1.583333333 + -1.083333333x + x2 = 0 Move the constant term to the right: Add '1.583333333' to each side of the equation. -1.583333333 + -1.083333333x + 1.583333333 + x2 = 0 + 1.583333333 Reorder the terms: -1.583333333 + 1.583333333 + -1.083333333x + x2 = 0 + 1.583333333 Combine like terms: -1.583333333 + 1.583333333 = 0.000000000 0.000000000 + -1.083333333x + x2 = 0 + 1.583333333 -1.083333333x + x2 = 0 + 1.583333333 Combine like terms: 0 + 1.583333333 = 1.583333333 -1.083333333x + x2 = 1.583333333 The x term is -1.083333333x. Take half its coefficient (-0.5416666665). Square it (0.2934027776) and add it to both sides. Add '0.2934027776' to each side of the equation. -1.083333333x + 0.2934027776 + x2 = 1.583333333 + 0.2934027776 Reorder the terms: 0.2934027776 + -1.083333333x + x2 = 1.583333333 + 0.2934027776 Combine like terms: 1.583333333 + 0.2934027776 = 1.8767361106 0.2934027776 + -1.083333333x + x2 = 1.8767361106 Factor a perfect square on the left side: (x + -0.5416666665)(x + -0.5416666665) = 1.8767361106 Calculate the square root of the right side: 1.369940185 Break this problem into two subproblems by setting (x + -0.5416666665) equal to 1.369940185 and -1.369940185.Subproblem 1
x + -0.5416666665 = 1.369940185 Simplifying x + -0.5416666665 = 1.369940185 Reorder the terms: -0.5416666665 + x = 1.369940185 Solving -0.5416666665 + x = 1.369940185 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + x = 1.369940185 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = 1.369940185 + 0.5416666665 x = 1.369940185 + 0.5416666665 Combine like terms: 1.369940185 + 0.5416666665 = 1.9116068515 x = 1.9116068515 Simplifying x = 1.9116068515Subproblem 2
x + -0.5416666665 = -1.369940185 Simplifying x + -0.5416666665 = -1.369940185 Reorder the terms: -0.5416666665 + x = -1.369940185 Solving -0.5416666665 + x = -1.369940185 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + x = -1.369940185 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = -1.369940185 + 0.5416666665 x = -1.369940185 + 0.5416666665 Combine like terms: -1.369940185 + 0.5416666665 = -0.8282735185 x = -0.8282735185 Simplifying x = -0.8282735185Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.9116068515, -0.8282735185}
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