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Simplifying (x + 3)(2x + -5) = 5 Reorder the terms: (3 + x)(2x + -5) = 5 Reorder the terms: (3 + x)(-5 + 2x) = 5 Multiply (3 + x) * (-5 + 2x) (3(-5 + 2x) + x(-5 + 2x)) = 5 ((-5 * 3 + 2x * 3) + x(-5 + 2x)) = 5 ((-15 + 6x) + x(-5 + 2x)) = 5 (-15 + 6x + (-5 * x + 2x * x)) = 5 (-15 + 6x + (-5x + 2x2)) = 5 Combine like terms: 6x + -5x = 1x (-15 + 1x + 2x2) = 5 Solving -15 + 1x + 2x2 = 5 Solving for variable 'x'. Reorder the terms: -15 + -5 + 1x + 2x2 = 5 + -5 Combine like terms: -15 + -5 = -20 -20 + 1x + 2x2 = 5 + -5 Combine like terms: 5 + -5 = 0 -20 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10 + 0.5x + x2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 0.5x + 10 + x2 = 0 + 10 Reorder the terms: -10 + 10 + 0.5x + x2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 0.5x + x2 = 0 + 10 0.5x + x2 = 0 + 10 Combine like terms: 0 + 10 = 10 0.5x + x2 = 10 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 = 10 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 10 + 0.0625 Combine like terms: 10 + 0.0625 = 10.0625 0.0625 + 0.5x + x2 = 10.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 10.0625 Calculate the square root of the right side: 3.172144385 Break this problem into two subproblems by setting (x + 0.25) equal to 3.172144385 and -3.172144385.Subproblem 1
x + 0.25 = 3.172144385 Simplifying x + 0.25 = 3.172144385 Reorder the terms: 0.25 + x = 3.172144385 Solving 0.25 + x = 3.172144385 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 = 3.172144385 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.172144385 + -0.25 x = 3.172144385 + -0.25 Combine like terms: 3.172144385 + -0.25 = 2.922144385 x = 2.922144385 Simplifying x = 2.922144385Subproblem 2
x + 0.25 = -3.172144385 Simplifying x + 0.25 = -3.172144385 Reorder the terms: 0.25 + x = -3.172144385 Solving 0.25 + x = -3.172144385 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 = -3.172144385 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.172144385 + -0.25 x = -3.172144385 + -0.25 Combine like terms: -3.172144385 + -0.25 = -3.422144385 x = -3.422144385 Simplifying x = -3.422144385Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.922144385, -3.422144385}
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