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Simplifying (2x + 3)(2x + 3) = 7 Reorder the terms: (3 + 2x)(2x + 3) = 7 Reorder the terms: (3 + 2x)(3 + 2x) = 7 Multiply (3 + 2x) * (3 + 2x) (3(3 + 2x) + 2x * (3 + 2x)) = 7 ((3 * 3 + 2x * 3) + 2x * (3 + 2x)) = 7 ((9 + 6x) + 2x * (3 + 2x)) = 7 (9 + 6x + (3 * 2x + 2x * 2x)) = 7 (9 + 6x + (6x + 4x2)) = 7 Combine like terms: 6x + 6x = 12x (9 + 12x + 4x2) = 7 Solving 9 + 12x + 4x2 = 7 Solving for variable 'x'. Reorder the terms: 9 + -7 + 12x + 4x2 = 7 + -7 Combine like terms: 9 + -7 = 2 2 + 12x + 4x2 = 7 + -7 Combine like terms: 7 + -7 = 0 2 + 12x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + 6x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + 6x + 2x2)' equal to zero and attempt to solve: Simplifying 1 + 6x + 2x2 = 0 Solving 1 + 6x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 3x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 3x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 3x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 3x + x2 = 0 + -0.5 3x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 3x + x2 = -0.5 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -0.5 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -0.5 + 2.25 Combine like terms: -0.5 + 2.25 = 1.75 2.25 + 3x + x2 = 1.75 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (x + 1.5) equal to 1.322875656 and -1.322875656.Subproblem 1
x + 1.5 = 1.322875656 Simplifying x + 1.5 = 1.322875656 Reorder the terms: 1.5 + x = 1.322875656 Solving 1.5 + x = 1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.322875656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.322875656 + -1.5 x = 1.322875656 + -1.5 Combine like terms: 1.322875656 + -1.5 = -0.177124344 x = -0.177124344 Simplifying x = -0.177124344Subproblem 2
x + 1.5 = -1.322875656 Simplifying x + 1.5 = -1.322875656 Reorder the terms: 1.5 + x = -1.322875656 Solving 1.5 + x = -1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.322875656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.322875656 + -1.5 x = -1.322875656 + -1.5 Combine like terms: -1.322875656 + -1.5 = -2.822875656 x = -2.822875656 Simplifying x = -2.822875656Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.177124344, -2.822875656}Solution
x = {-0.177124344, -2.822875656}
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