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Simplifying (y + 7) * y = 5 Reorder the terms: (7 + y) * y = 5 Reorder the terms for easier multiplication: y(7 + y) = 5 (7 * y + y * y) = 5 (7y + y2) = 5 Solving 7y + y2 = 5 Solving for variable 'y'. Reorder the terms: -5 + 7y + y2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 7y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 7y + 5 + y2 = 0 + 5 Reorder the terms: -5 + 5 + 7y + y2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 7y + y2 = 0 + 5 7y + y2 = 0 + 5 Combine like terms: 0 + 5 = 5 7y + y2 = 5 The y term is 7y. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7y + 12.25 + y2 = 5 + 12.25 Reorder the terms: 12.25 + 7y + y2 = 5 + 12.25 Combine like terms: 5 + 12.25 = 17.25 12.25 + 7y + y2 = 17.25 Factor a perfect square on the left side: (y + 3.5)(y + 3.5) = 17.25 Calculate the square root of the right side: 4.153311931 Break this problem into two subproblems by setting (y + 3.5) equal to 4.153311931 and -4.153311931.Subproblem 1
y + 3.5 = 4.153311931 Simplifying y + 3.5 = 4.153311931 Reorder the terms: 3.5 + y = 4.153311931 Solving 3.5 + y = 4.153311931 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = 4.153311931 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = 4.153311931 + -3.5 y = 4.153311931 + -3.5 Combine like terms: 4.153311931 + -3.5 = 0.653311931 y = 0.653311931 Simplifying y = 0.653311931Subproblem 2
y + 3.5 = -4.153311931 Simplifying y + 3.5 = -4.153311931 Reorder the terms: 3.5 + y = -4.153311931 Solving 3.5 + y = -4.153311931 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = -4.153311931 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = -4.153311931 + -3.5 y = -4.153311931 + -3.5 Combine like terms: -4.153311931 + -3.5 = -7.653311931 y = -7.653311931 Simplifying y = -7.653311931Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.653311931, -7.653311931}
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