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Simplifying (x2 + 5)(x2 + 5) + -15(x2 + 5) + 53 = 0 Reorder the terms: (5 + x2)(x2 + 5) + -15(x2 + 5) + 53 = 0 Reorder the terms: (5 + x2)(5 + x2) + -15(x2 + 5) + 53 = 0 Multiply (5 + x2) * (5 + x2) (5(5 + x2) + x2(5 + x2)) + -15(x2 + 5) + 53 = 0 ((5 * 5 + x2 * 5) + x2(5 + x2)) + -15(x2 + 5) + 53 = 0 ((25 + 5x2) + x2(5 + x2)) + -15(x2 + 5) + 53 = 0 (25 + 5x2 + (5 * x2 + x2 * x2)) + -15(x2 + 5) + 53 = 0 (25 + 5x2 + (5x2 + x4)) + -15(x2 + 5) + 53 = 0 Combine like terms: 5x2 + 5x2 = 10x2 (25 + 10x2 + x4) + -15(x2 + 5) + 53 = 0 Reorder the terms: 25 + 10x2 + x4 + -15(5 + x2) + 53 = 0 25 + 10x2 + x4 + (5 * -15 + x2 * -15) + 53 = 0 25 + 10x2 + x4 + (-75 + -15x2) + 53 = 0 Reorder the terms: 25 + -75 + 53 + 10x2 + -15x2 + x4 = 0 Combine like terms: 25 + -75 = -50 -50 + 53 + 10x2 + -15x2 + x4 = 0 Combine like terms: -50 + 53 = 3 3 + 10x2 + -15x2 + x4 = 0 Combine like terms: 10x2 + -15x2 = -5x2 3 + -5x2 + x4 = 0 Solving 3 + -5x2 + x4 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -5x2 + -3 + x4 = 0 + -3 Reorder the terms: 3 + -3 + -5x2 + x4 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -5x2 + x4 = 0 + -3 -5x2 + x4 = 0 + -3 Combine like terms: 0 + -3 = -3 -5x2 + x4 = -3 The x term is -5x2. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x2 + 6.25 + x4 = -3 + 6.25 Reorder the terms: 6.25 + -5x2 + x4 = -3 + 6.25 Combine like terms: -3 + 6.25 = 3.25 6.25 + -5x2 + x4 = 3.25 Factor a perfect square on the left side: (x2 + -2.5)(x2 + -2.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (x2 + -2.5) equal to 1.802775638 and -1.802775638.Subproblem 1
x2 + -2.5 = 1.802775638 Simplifying x2 + -2.5 = 1.802775638 Reorder the terms: -2.5 + x2 = 1.802775638 Solving -2.5 + x2 = 1.802775638 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x2 = 1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x2 = 1.802775638 + 2.5 x2 = 1.802775638 + 2.5 Combine like terms: 1.802775638 + 2.5 = 4.302775638 x2 = 4.302775638 Simplifying x2 = 4.302775638 Take the square root of each side: x = {-2.074313293, 2.074313293}Subproblem 2
x2 + -2.5 = -1.802775638 Simplifying x2 + -2.5 = -1.802775638 Reorder the terms: -2.5 + x2 = -1.802775638 Solving -2.5 + x2 = -1.802775638 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x2 = -1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x2 = -1.802775638 + 2.5 x2 = -1.802775638 + 2.5 Combine like terms: -1.802775638 + 2.5 = 0.697224362 x2 = 0.697224362 Simplifying x2 = 0.697224362 Take the square root of each side: x = {-0.834999618, 0.834999618}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.074313293, 2.074313293, -0.834999618, 0.834999618}
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