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Simplifying (3x + -4)(2x + 3) = 78 Reorder the terms: (-4 + 3x)(2x + 3) = 78 Reorder the terms: (-4 + 3x)(3 + 2x) = 78 Multiply (-4 + 3x) * (3 + 2x) (-4(3 + 2x) + 3x * (3 + 2x)) = 78 ((3 * -4 + 2x * -4) + 3x * (3 + 2x)) = 78 ((-12 + -8x) + 3x * (3 + 2x)) = 78 (-12 + -8x + (3 * 3x + 2x * 3x)) = 78 (-12 + -8x + (9x + 6x2)) = 78 Combine like terms: -8x + 9x = 1x (-12 + 1x + 6x2) = 78 Solving -12 + 1x + 6x2 = 78 Solving for variable 'x'. Reorder the terms: -12 + -78 + 1x + 6x2 = 78 + -78 Combine like terms: -12 + -78 = -90 -90 + 1x + 6x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -90 + 1x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -15 + 0.1666666667x + x2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 0.1666666667x + 15 + x2 = 0 + 15 Reorder the terms: -15 + 15 + 0.1666666667x + x2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 0.1666666667x + x2 = 0 + 15 0.1666666667x + x2 = 0 + 15 Combine like terms: 0 + 15 = 15 0.1666666667x + x2 = 15 The x term is 0.1666666667x. Take half its coefficient (0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. 0.1666666667x + 0.006944444447 + x2 = 15 + 0.006944444447 Reorder the terms: 0.006944444447 + 0.1666666667x + x2 = 15 + 0.006944444447 Combine like terms: 15 + 0.006944444447 = 15.006944444447 0.006944444447 + 0.1666666667x + x2 = 15.006944444447 Factor a perfect square on the left side: (x + 0.08333333335)(x + 0.08333333335) = 15.006944444447 Calculate the square root of the right side: 3.873879766 Break this problem into two subproblems by setting (x + 0.08333333335) equal to 3.873879766 and -3.873879766.Subproblem 1
x + 0.08333333335 = 3.873879766 Simplifying x + 0.08333333335 = 3.873879766 Reorder the terms: 0.08333333335 + x = 3.873879766 Solving 0.08333333335 + x = 3.873879766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 3.873879766 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 3.873879766 + -0.08333333335 x = 3.873879766 + -0.08333333335 Combine like terms: 3.873879766 + -0.08333333335 = 3.79054643265 x = 3.79054643265 Simplifying x = 3.79054643265Subproblem 2
x + 0.08333333335 = -3.873879766 Simplifying x + 0.08333333335 = -3.873879766 Reorder the terms: 0.08333333335 + x = -3.873879766 Solving 0.08333333335 + x = -3.873879766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -3.873879766 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -3.873879766 + -0.08333333335 x = -3.873879766 + -0.08333333335 Combine like terms: -3.873879766 + -0.08333333335 = -3.95721309935 x = -3.95721309935 Simplifying x = -3.95721309935Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.79054643265, -3.95721309935}
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