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Simplifying 3x(x + 5) + 4x = 78 Reorder the terms: 3x(5 + x) + 4x = 78 (5 * 3x + x * 3x) + 4x = 78 (15x + 3x2) + 4x = 78 Reorder the terms: 15x + 4x + 3x2 = 78 Combine like terms: 15x + 4x = 19x 19x + 3x2 = 78 Solving 19x + 3x2 = 78 Solving for variable 'x'. Reorder the terms: -78 + 19x + 3x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -78 + 19x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -26 + 6.333333333x + x2 = 0 Move the constant term to the right: Add '26' to each side of the equation. -26 + 6.333333333x + 26 + x2 = 0 + 26 Reorder the terms: -26 + 26 + 6.333333333x + x2 = 0 + 26 Combine like terms: -26 + 26 = 0 0 + 6.333333333x + x2 = 0 + 26 6.333333333x + x2 = 0 + 26 Combine like terms: 0 + 26 = 26 6.333333333x + x2 = 26 The x term is 6.333333333x. Take half its coefficient (3.166666667). Square it (10.02777778) and add it to both sides. Add '10.02777778' to each side of the equation. 6.333333333x + 10.02777778 + x2 = 26 + 10.02777778 Reorder the terms: 10.02777778 + 6.333333333x + x2 = 26 + 10.02777778 Combine like terms: 26 + 10.02777778 = 36.02777778 10.02777778 + 6.333333333x + x2 = 36.02777778 Factor a perfect square on the left side: (x + 3.166666667)(x + 3.166666667) = 36.02777778 Calculate the square root of the right side: 6.002314369 Break this problem into two subproblems by setting (x + 3.166666667) equal to 6.002314369 and -6.002314369.Subproblem 1
x + 3.166666667 = 6.002314369 Simplifying x + 3.166666667 = 6.002314369 Reorder the terms: 3.166666667 + x = 6.002314369 Solving 3.166666667 + x = 6.002314369 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + x = 6.002314369 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = 6.002314369 + -3.166666667 x = 6.002314369 + -3.166666667 Combine like terms: 6.002314369 + -3.166666667 = 2.835647702 x = 2.835647702 Simplifying x = 2.835647702Subproblem 2
x + 3.166666667 = -6.002314369 Simplifying x + 3.166666667 = -6.002314369 Reorder the terms: 3.166666667 + x = -6.002314369 Solving 3.166666667 + x = -6.002314369 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + x = -6.002314369 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = -6.002314369 + -3.166666667 x = -6.002314369 + -3.166666667 Combine like terms: -6.002314369 + -3.166666667 = -9.168981036 x = -9.168981036 Simplifying x = -9.168981036Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.835647702, -9.168981036}
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