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Simplifying 5(y2 + 3) + 7y(y + -13) = 0 Reorder the terms: 5(3 + y2) + 7y(y + -13) = 0 (3 * 5 + y2 * 5) + 7y(y + -13) = 0 (15 + 5y2) + 7y(y + -13) = 0 Reorder the terms: 15 + 5y2 + 7y(-13 + y) = 0 15 + 5y2 + (-13 * 7y + y * 7y) = 0 15 + 5y2 + (-91y + 7y2) = 0 Reorder the terms: 15 + -91y + 5y2 + 7y2 = 0 Combine like terms: 5y2 + 7y2 = 12y2 15 + -91y + 12y2 = 0 Solving 15 + -91y + 12y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 1.25 + -7.583333333y + y2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + -7.583333333y + -1.25 + y2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + -7.583333333y + y2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + -7.583333333y + y2 = 0 + -1.25 -7.583333333y + y2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 -7.583333333y + y2 = -1.25 The y term is -7.583333333y. Take half its coefficient (-3.791666667). Square it (14.37673611) and add it to both sides. Add '14.37673611' to each side of the equation. -7.583333333y + 14.37673611 + y2 = -1.25 + 14.37673611 Reorder the terms: 14.37673611 + -7.583333333y + y2 = -1.25 + 14.37673611 Combine like terms: -1.25 + 14.37673611 = 13.12673611 14.37673611 + -7.583333333y + y2 = 13.12673611 Factor a perfect square on the left side: (y + -3.791666667)(y + -3.791666667) = 13.12673611 Calculate the square root of the right side: 3.623083785 Break this problem into two subproblems by setting (y + -3.791666667) equal to 3.623083785 and -3.623083785.Subproblem 1
y + -3.791666667 = 3.623083785 Simplifying y + -3.791666667 = 3.623083785 Reorder the terms: -3.791666667 + y = 3.623083785 Solving -3.791666667 + y = 3.623083785 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.791666667' to each side of the equation. -3.791666667 + 3.791666667 + y = 3.623083785 + 3.791666667 Combine like terms: -3.791666667 + 3.791666667 = 0.000000000 0.000000000 + y = 3.623083785 + 3.791666667 y = 3.623083785 + 3.791666667 Combine like terms: 3.623083785 + 3.791666667 = 7.414750452 y = 7.414750452 Simplifying y = 7.414750452Subproblem 2
y + -3.791666667 = -3.623083785 Simplifying y + -3.791666667 = -3.623083785 Reorder the terms: -3.791666667 + y = -3.623083785 Solving -3.791666667 + y = -3.623083785 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.791666667' to each side of the equation. -3.791666667 + 3.791666667 + y = -3.623083785 + 3.791666667 Combine like terms: -3.791666667 + 3.791666667 = 0.000000000 0.000000000 + y = -3.623083785 + 3.791666667 y = -3.623083785 + 3.791666667 Combine like terms: -3.623083785 + 3.791666667 = 0.168582882 y = 0.168582882 Simplifying y = 0.168582882Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.414750452, 0.168582882}
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