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Simplifying 3n2 + -22n + 7 = 4 Reorder the terms: 7 + -22n + 3n2 = 4 Solving 7 + -22n + 3n2 = 4 Solving for variable 'n'. Reorder the terms: 7 + -4 + -22n + 3n2 = 4 + -4 Combine like terms: 7 + -4 = 3 3 + -22n + 3n2 = 4 + -4 Combine like terms: 4 + -4 = 0 3 + -22n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1 + -7.333333333n + n2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -7.333333333n + -1 + n2 = 0 + -1 Reorder the terms: 1 + -1 + -7.333333333n + n2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -7.333333333n + n2 = 0 + -1 -7.333333333n + n2 = 0 + -1 Combine like terms: 0 + -1 = -1 -7.333333333n + n2 = -1 The n term is -7.333333333n. Take half its coefficient (-3.666666667). Square it (13.44444445) and add it to both sides. Add '13.44444445' to each side of the equation. -7.333333333n + 13.44444445 + n2 = -1 + 13.44444445 Reorder the terms: 13.44444445 + -7.333333333n + n2 = -1 + 13.44444445 Combine like terms: -1 + 13.44444445 = 12.44444445 13.44444445 + -7.333333333n + n2 = 12.44444445 Factor a perfect square on the left side: (n + -3.666666667)(n + -3.666666667) = 12.44444445 Calculate the square root of the right side: 3.527668416 Break this problem into two subproblems by setting (n + -3.666666667) equal to 3.527668416 and -3.527668416.Subproblem 1
n + -3.666666667 = 3.527668416 Simplifying n + -3.666666667 = 3.527668416 Reorder the terms: -3.666666667 + n = 3.527668416 Solving -3.666666667 + n = 3.527668416 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.666666667' to each side of the equation. -3.666666667 + 3.666666667 + n = 3.527668416 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + n = 3.527668416 + 3.666666667 n = 3.527668416 + 3.666666667 Combine like terms: 3.527668416 + 3.666666667 = 7.194335083 n = 7.194335083 Simplifying n = 7.194335083Subproblem 2
n + -3.666666667 = -3.527668416 Simplifying n + -3.666666667 = -3.527668416 Reorder the terms: -3.666666667 + n = -3.527668416 Solving -3.666666667 + n = -3.527668416 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.666666667' to each side of the equation. -3.666666667 + 3.666666667 + n = -3.527668416 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + n = -3.527668416 + 3.666666667 n = -3.527668416 + 3.666666667 Combine like terms: -3.527668416 + 3.666666667 = 0.138998251 n = 0.138998251 Simplifying n = 0.138998251Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.194335083, 0.138998251}
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