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Simplifying (4x + 1)(7x + 3) = 65 Reorder the terms: (1 + 4x)(7x + 3) = 65 Reorder the terms: (1 + 4x)(3 + 7x) = 65 Multiply (1 + 4x) * (3 + 7x) (1(3 + 7x) + 4x * (3 + 7x)) = 65 ((3 * 1 + 7x * 1) + 4x * (3 + 7x)) = 65 ((3 + 7x) + 4x * (3 + 7x)) = 65 (3 + 7x + (3 * 4x + 7x * 4x)) = 65 (3 + 7x + (12x + 28x2)) = 65 Combine like terms: 7x + 12x = 19x (3 + 19x + 28x2) = 65 Solving 3 + 19x + 28x2 = 65 Solving for variable 'x'. Reorder the terms: 3 + -65 + 19x + 28x2 = 65 + -65 Combine like terms: 3 + -65 = -62 -62 + 19x + 28x2 = 65 + -65 Combine like terms: 65 + -65 = 0 -62 + 19x + 28x2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -2.214285714 + 0.6785714286x + x2 = 0 Move the constant term to the right: Add '2.214285714' to each side of the equation. -2.214285714 + 0.6785714286x + 2.214285714 + x2 = 0 + 2.214285714 Reorder the terms: -2.214285714 + 2.214285714 + 0.6785714286x + x2 = 0 + 2.214285714 Combine like terms: -2.214285714 + 2.214285714 = 0.000000000 0.000000000 + 0.6785714286x + x2 = 0 + 2.214285714 0.6785714286x + x2 = 0 + 2.214285714 Combine like terms: 0 + 2.214285714 = 2.214285714 0.6785714286x + x2 = 2.214285714 The x term is 0.6785714286x. Take half its coefficient (0.3392857143). Square it (0.1151147959) and add it to both sides. Add '0.1151147959' to each side of the equation. 0.6785714286x + 0.1151147959 + x2 = 2.214285714 + 0.1151147959 Reorder the terms: 0.1151147959 + 0.6785714286x + x2 = 2.214285714 + 0.1151147959 Combine like terms: 2.214285714 + 0.1151147959 = 2.3294005099 0.1151147959 + 0.6785714286x + x2 = 2.3294005099 Factor a perfect square on the left side: (x + 0.3392857143)(x + 0.3392857143) = 2.3294005099 Calculate the square root of the right side: 1.52623737 Break this problem into two subproblems by setting (x + 0.3392857143) equal to 1.52623737 and -1.52623737.Subproblem 1
x + 0.3392857143 = 1.52623737 Simplifying x + 0.3392857143 = 1.52623737 Reorder the terms: 0.3392857143 + x = 1.52623737 Solving 0.3392857143 + x = 1.52623737 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3392857143' to each side of the equation. 0.3392857143 + -0.3392857143 + x = 1.52623737 + -0.3392857143 Combine like terms: 0.3392857143 + -0.3392857143 = 0.0000000000 0.0000000000 + x = 1.52623737 + -0.3392857143 x = 1.52623737 + -0.3392857143 Combine like terms: 1.52623737 + -0.3392857143 = 1.1869516557 x = 1.1869516557 Simplifying x = 1.1869516557Subproblem 2
x + 0.3392857143 = -1.52623737 Simplifying x + 0.3392857143 = -1.52623737 Reorder the terms: 0.3392857143 + x = -1.52623737 Solving 0.3392857143 + x = -1.52623737 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3392857143' to each side of the equation. 0.3392857143 + -0.3392857143 + x = -1.52623737 + -0.3392857143 Combine like terms: 0.3392857143 + -0.3392857143 = 0.0000000000 0.0000000000 + x = -1.52623737 + -0.3392857143 x = -1.52623737 + -0.3392857143 Combine like terms: -1.52623737 + -0.3392857143 = -1.8655230843 x = -1.8655230843 Simplifying x = -1.8655230843Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.1869516557, -1.8655230843}
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