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Simplifying (3x + 4)(5x + -7) = (2x + 7) + 53 Reorder the terms: (4 + 3x)(5x + -7) = (2x + 7) + 53 Reorder the terms: (4 + 3x)(-7 + 5x) = (2x + 7) + 53 Multiply (4 + 3x) * (-7 + 5x) (4(-7 + 5x) + 3x * (-7 + 5x)) = (2x + 7) + 53 ((-7 * 4 + 5x * 4) + 3x * (-7 + 5x)) = (2x + 7) + 53 ((-28 + 20x) + 3x * (-7 + 5x)) = (2x + 7) + 53 (-28 + 20x + (-7 * 3x + 5x * 3x)) = (2x + 7) + 53 (-28 + 20x + (-21x + 15x2)) = (2x + 7) + 53 Combine like terms: 20x + -21x = -1x (-28 + -1x + 15x2) = (2x + 7) + 53 Reorder the terms: -28 + -1x + 15x2 = (7 + 2x) + 53 Remove parenthesis around (7 + 2x) -28 + -1x + 15x2 = 7 + 2x + 53 Reorder the terms: -28 + -1x + 15x2 = 7 + 53 + 2x Combine like terms: 7 + 53 = 60 -28 + -1x + 15x2 = 60 + 2x Solving -28 + -1x + 15x2 = 60 + 2x Solving for variable 'x'. Reorder the terms: -28 + -60 + -1x + -2x + 15x2 = 60 + 2x + -60 + -2x Combine like terms: -28 + -60 = -88 -88 + -1x + -2x + 15x2 = 60 + 2x + -60 + -2x Combine like terms: -1x + -2x = -3x -88 + -3x + 15x2 = 60 + 2x + -60 + -2x Reorder the terms: -88 + -3x + 15x2 = 60 + -60 + 2x + -2x Combine like terms: 60 + -60 = 0 -88 + -3x + 15x2 = 0 + 2x + -2x -88 + -3x + 15x2 = 2x + -2x Combine like terms: 2x + -2x = 0 -88 + -3x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -5.866666667 + -0.2x + x2 = 0 Move the constant term to the right: Add '5.866666667' to each side of the equation. -5.866666667 + -0.2x + 5.866666667 + x2 = 0 + 5.866666667 Reorder the terms: -5.866666667 + 5.866666667 + -0.2x + x2 = 0 + 5.866666667 Combine like terms: -5.866666667 + 5.866666667 = 0.000000000 0.000000000 + -0.2x + x2 = 0 + 5.866666667 -0.2x + x2 = 0 + 5.866666667 Combine like terms: 0 + 5.866666667 = 5.866666667 -0.2x + x2 = 5.866666667 The x term is -0.2x. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2x + 0.01 + x2 = 5.866666667 + 0.01 Reorder the terms: 0.01 + -0.2x + x2 = 5.866666667 + 0.01 Combine like terms: 5.866666667 + 0.01 = 5.876666667 0.01 + -0.2x + x2 = 5.876666667 Factor a perfect square on the left side: (x + -0.1)(x + -0.1) = 5.876666667 Calculate the square root of the right side: 2.424183711 Break this problem into two subproblems by setting (x + -0.1) equal to 2.424183711 and -2.424183711.Subproblem 1
x + -0.1 = 2.424183711 Simplifying x + -0.1 = 2.424183711 Reorder the terms: -0.1 + x = 2.424183711 Solving -0.1 + x = 2.424183711 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 2.424183711 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 2.424183711 + 0.1 x = 2.424183711 + 0.1 Combine like terms: 2.424183711 + 0.1 = 2.524183711 x = 2.524183711 Simplifying x = 2.524183711Subproblem 2
x + -0.1 = -2.424183711 Simplifying x + -0.1 = -2.424183711 Reorder the terms: -0.1 + x = -2.424183711 Solving -0.1 + x = -2.424183711 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -2.424183711 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -2.424183711 + 0.1 x = -2.424183711 + 0.1 Combine like terms: -2.424183711 + 0.1 = -2.324183711 x = -2.324183711 Simplifying x = -2.324183711Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.524183711, -2.324183711}
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