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Simplifying (2X + -1)(2X + 1) + -4(X + 5) = 19 Reorder the terms: (-1 + 2X)(2X + 1) + -4(X + 5) = 19 Reorder the terms: (-1 + 2X)(1 + 2X) + -4(X + 5) = 19 Multiply (-1 + 2X) * (1 + 2X) (-1(1 + 2X) + 2X * (1 + 2X)) + -4(X + 5) = 19 ((1 * -1 + 2X * -1) + 2X * (1 + 2X)) + -4(X + 5) = 19 ((-1 + -2X) + 2X * (1 + 2X)) + -4(X + 5) = 19 (-1 + -2X + (1 * 2X + 2X * 2X)) + -4(X + 5) = 19 (-1 + -2X + (2X + 4X2)) + -4(X + 5) = 19 Combine like terms: -2X + 2X = 0 (-1 + 0 + 4X2) + -4(X + 5) = 19 (-1 + 4X2) + -4(X + 5) = 19 Reorder the terms: -1 + 4X2 + -4(5 + X) = 19 -1 + 4X2 + (5 * -4 + X * -4) = 19 -1 + 4X2 + (-20 + -4X) = 19 Reorder the terms: -1 + -20 + -4X + 4X2 = 19 Combine like terms: -1 + -20 = -21 -21 + -4X + 4X2 = 19 Solving -21 + -4X + 4X2 = 19 Solving for variable 'X'. Reorder the terms: -21 + -19 + -4X + 4X2 = 19 + -19 Combine like terms: -21 + -19 = -40 -40 + -4X + 4X2 = 19 + -19 Combine like terms: 19 + -19 = 0 -40 + -4X + 4X2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-10 + -1X + X2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-10 + -1X + X2)' equal to zero and attempt to solve: Simplifying -10 + -1X + X2 = 0 Solving -10 + -1X + X2 = 0 Begin completing the square. Move the constant term to the right: Add '10' to each side of the equation. -10 + -1X + 10 + X2 = 0 + 10 Reorder the terms: -10 + 10 + -1X + X2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -1X + X2 = 0 + 10 -1X + X2 = 0 + 10 Combine like terms: 0 + 10 = 10 -1X + X2 = 10 The X term is -1X. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1X + 0.25 + X2 = 10 + 0.25 Reorder the terms: 0.25 + -1X + X2 = 10 + 0.25 Combine like terms: 10 + 0.25 = 10.25 0.25 + -1X + X2 = 10.25 Factor a perfect square on the left side: (X + -0.5)(X + -0.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (X + -0.5) equal to 3.201562119 and -3.201562119.Subproblem 1
X + -0.5 = 3.201562119 Simplifying X + -0.5 = 3.201562119 Reorder the terms: -0.5 + X = 3.201562119 Solving -0.5 + X = 3.201562119 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + X = 3.201562119 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + X = 3.201562119 + 0.5 X = 3.201562119 + 0.5 Combine like terms: 3.201562119 + 0.5 = 3.701562119 X = 3.701562119 Simplifying X = 3.701562119Subproblem 2
X + -0.5 = -3.201562119 Simplifying X + -0.5 = -3.201562119 Reorder the terms: -0.5 + X = -3.201562119 Solving -0.5 + X = -3.201562119 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + X = -3.201562119 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + X = -3.201562119 + 0.5 X = -3.201562119 + 0.5 Combine like terms: -3.201562119 + 0.5 = -2.701562119 X = -2.701562119 Simplifying X = -2.701562119Solution
The solution to the problem is based on the solutions from the subproblems. X = {3.701562119, -2.701562119}Solution
X = {3.701562119, -2.701562119}
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