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Simplifying (x + 3)(2x + -5) = 38 Reorder the terms: (3 + x)(2x + -5) = 38 Reorder the terms: (3 + x)(-5 + 2x) = 38 Multiply (3 + x) * (-5 + 2x) (3(-5 + 2x) + x(-5 + 2x)) = 38 ((-5 * 3 + 2x * 3) + x(-5 + 2x)) = 38 ((-15 + 6x) + x(-5 + 2x)) = 38 (-15 + 6x + (-5 * x + 2x * x)) = 38 (-15 + 6x + (-5x + 2x2)) = 38 Combine like terms: 6x + -5x = 1x (-15 + 1x + 2x2) = 38 Solving -15 + 1x + 2x2 = 38 Solving for variable 'x'. Reorder the terms: -15 + -38 + 1x + 2x2 = 38 + -38 Combine like terms: -15 + -38 = -53 -53 + 1x + 2x2 = 38 + -38 Combine like terms: 38 + -38 = 0 -53 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -26.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '26.5' to each side of the equation. -26.5 + 0.5x + 26.5 + x2 = 0 + 26.5 Reorder the terms: -26.5 + 26.5 + 0.5x + x2 = 0 + 26.5 Combine like terms: -26.5 + 26.5 = 0.0 0.0 + 0.5x + x2 = 0 + 26.5 0.5x + x2 = 0 + 26.5 Combine like terms: 0 + 26.5 = 26.5 0.5x + x2 = 26.5 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 26.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 26.5 + 0.0625 Combine like terms: 26.5 + 0.0625 = 26.5625 0.0625 + 0.5x + x2 = 26.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 26.5625 Calculate the square root of the right side: 5.153882032 Break this problem into two subproblems by setting (x + 0.25) equal to 5.153882032 and -5.153882032.Subproblem 1
x + 0.25 = 5.153882032 Simplifying x + 0.25 = 5.153882032 Reorder the terms: 0.25 + x = 5.153882032 Solving 0.25 + x = 5.153882032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 5.153882032 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.153882032 + -0.25 x = 5.153882032 + -0.25 Combine like terms: 5.153882032 + -0.25 = 4.903882032 x = 4.903882032 Simplifying x = 4.903882032Subproblem 2
x + 0.25 = -5.153882032 Simplifying x + 0.25 = -5.153882032 Reorder the terms: 0.25 + x = -5.153882032 Solving 0.25 + x = -5.153882032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -5.153882032 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.153882032 + -0.25 x = -5.153882032 + -0.25 Combine like terms: -5.153882032 + -0.25 = -5.403882032 x = -5.403882032 Simplifying x = -5.403882032Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.903882032, -5.403882032}
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