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Simplifying 3(2x + -7)(5x + 12) = -14 Reorder the terms: 3(-7 + 2x)(5x + 12) = -14 Reorder the terms: 3(-7 + 2x)(12 + 5x) = -14 Multiply (-7 + 2x) * (12 + 5x) 3(-7(12 + 5x) + 2x * (12 + 5x)) = -14 3((12 * -7 + 5x * -7) + 2x * (12 + 5x)) = -14 3((-84 + -35x) + 2x * (12 + 5x)) = -14 3(-84 + -35x + (12 * 2x + 5x * 2x)) = -14 3(-84 + -35x + (24x + 10x2)) = -14 Combine like terms: -35x + 24x = -11x 3(-84 + -11x + 10x2) = -14 (-84 * 3 + -11x * 3 + 10x2 * 3) = -14 (-252 + -33x + 30x2) = -14 Solving -252 + -33x + 30x2 = -14 Solving for variable 'x'. Reorder the terms: -252 + 14 + -33x + 30x2 = -14 + 14 Combine like terms: -252 + 14 = -238 -238 + -33x + 30x2 = -14 + 14 Combine like terms: -14 + 14 = 0 -238 + -33x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -7.933333333 + -1.1x + x2 = 0.0 Move the constant term to the right: Add '7.933333333' to each side of the equation. -7.933333333 + -1.1x + 7.933333333 + x2 = 0.0 + 7.933333333 Reorder the terms: -7.933333333 + 7.933333333 + -1.1x + x2 = 0.0 + 7.933333333 Combine like terms: -7.933333333 + 7.933333333 = 0.000000000 0.000000000 + -1.1x + x2 = 0.0 + 7.933333333 -1.1x + x2 = 0.0 + 7.933333333 Combine like terms: 0.0 + 7.933333333 = 7.933333333 -1.1x + x2 = 7.933333333 The x term is -1.1x. Take half its coefficient (-0.55). Square it (0.3025) and add it to both sides. Add '0.3025' to each side of the equation. -1.1x + 0.3025 + x2 = 7.933333333 + 0.3025 Reorder the terms: 0.3025 + -1.1x + x2 = 7.933333333 + 0.3025 Combine like terms: 7.933333333 + 0.3025 = 8.235833333 0.3025 + -1.1x + x2 = 8.235833333 Factor a perfect square on the left side: (x + -0.55)(x + -0.55) = 8.235833333 Calculate the square root of the right side: 2.869814163 Break this problem into two subproblems by setting (x + -0.55) equal to 2.869814163 and -2.869814163.Subproblem 1
x + -0.55 = 2.869814163 Simplifying x + -0.55 = 2.869814163 Reorder the terms: -0.55 + x = 2.869814163 Solving -0.55 + x = 2.869814163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.55' to each side of the equation. -0.55 + 0.55 + x = 2.869814163 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + x = 2.869814163 + 0.55 x = 2.869814163 + 0.55 Combine like terms: 2.869814163 + 0.55 = 3.419814163 x = 3.419814163 Simplifying x = 3.419814163Subproblem 2
x + -0.55 = -2.869814163 Simplifying x + -0.55 = -2.869814163 Reorder the terms: -0.55 + x = -2.869814163 Solving -0.55 + x = -2.869814163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.55' to each side of the equation. -0.55 + 0.55 + x = -2.869814163 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + x = -2.869814163 + 0.55 x = -2.869814163 + 0.55 Combine like terms: -2.869814163 + 0.55 = -2.319814163 x = -2.319814163 Simplifying x = -2.319814163Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.419814163, -2.319814163}
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