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Simplifying 0 = 4y2 + -15y + 13 Reorder the terms: 0 = 13 + -15y + 4y2 Solving 0 = 13 + -15y + 4y2 Solving for variable 'y'. Combine like terms: 0 + -13 = -13 -13 + 15y + -4y2 = 13 + -15y + 4y2 + -13 + 15y + -4y2 Reorder the terms: -13 + 15y + -4y2 = 13 + -13 + -15y + 15y + 4y2 + -4y2 Combine like terms: 13 + -13 = 0 -13 + 15y + -4y2 = 0 + -15y + 15y + 4y2 + -4y2 -13 + 15y + -4y2 = -15y + 15y + 4y2 + -4y2 Combine like terms: -15y + 15y = 0 -13 + 15y + -4y2 = 0 + 4y2 + -4y2 -13 + 15y + -4y2 = 4y2 + -4y2 Combine like terms: 4y2 + -4y2 = 0 -13 + 15y + -4y2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 3.25 + -3.75y + y2 = 0 Move the constant term to the right: Add '-3.25' to each side of the equation. 3.25 + -3.75y + -3.25 + y2 = 0 + -3.25 Reorder the terms: 3.25 + -3.25 + -3.75y + y2 = 0 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + -3.75y + y2 = 0 + -3.25 -3.75y + y2 = 0 + -3.25 Combine like terms: 0 + -3.25 = -3.25 -3.75y + y2 = -3.25 The y term is -3.75y. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75y + 3.515625 + y2 = -3.25 + 3.515625 Reorder the terms: 3.515625 + -3.75y + y2 = -3.25 + 3.515625 Combine like terms: -3.25 + 3.515625 = 0.265625 3.515625 + -3.75y + y2 = 0.265625 Factor a perfect square on the left side: (y + -1.875)(y + -1.875) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (y + -1.875) equal to 0.515388203 and -0.515388203.Subproblem 1
y + -1.875 = 0.515388203 Simplifying y + -1.875 = 0.515388203 Reorder the terms: -1.875 + y = 0.515388203 Solving -1.875 + y = 0.515388203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + y = 0.515388203 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + y = 0.515388203 + 1.875 y = 0.515388203 + 1.875 Combine like terms: 0.515388203 + 1.875 = 2.390388203 y = 2.390388203 Simplifying y = 2.390388203Subproblem 2
y + -1.875 = -0.515388203 Simplifying y + -1.875 = -0.515388203 Reorder the terms: -1.875 + y = -0.515388203 Solving -1.875 + y = -0.515388203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + y = -0.515388203 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + y = -0.515388203 + 1.875 y = -0.515388203 + 1.875 Combine like terms: -0.515388203 + 1.875 = 1.359611797 y = 1.359611797 Simplifying y = 1.359611797Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.390388203, 1.359611797}
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