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Simplifying 15a2 + -3a = 3 + 7a Reorder the terms: -3a + 15a2 = 3 + 7a Solving -3a + 15a2 = 3 + 7a Solving for variable 'a'. Reorder the terms: -3 + -3a + -7a + 15a2 = 3 + 7a + -3 + -7a Combine like terms: -3a + -7a = -10a -3 + -10a + 15a2 = 3 + 7a + -3 + -7a Reorder the terms: -3 + -10a + 15a2 = 3 + -3 + 7a + -7a Combine like terms: 3 + -3 = 0 -3 + -10a + 15a2 = 0 + 7a + -7a -3 + -10a + 15a2 = 7a + -7a Combine like terms: 7a + -7a = 0 -3 + -10a + 15a2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.2 + -0.6666666667a + a2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.6666666667a + 0.2 + a2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.6666666667a + a2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.6666666667a + a2 = 0 + 0.2 -0.6666666667a + a2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -0.6666666667a + a2 = 0.2 The a term is -0.6666666667a. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667a + 0.1111111112 + a2 = 0.2 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667a + a2 = 0.2 + 0.1111111112 Combine like terms: 0.2 + 0.1111111112 = 0.3111111112 0.1111111112 + -0.6666666667a + a2 = 0.3111111112 Factor a perfect square on the left side: (a + -0.3333333334)(a + -0.3333333334) = 0.3111111112 Calculate the square root of the right side: 0.557773351 Break this problem into two subproblems by setting (a + -0.3333333334) equal to 0.557773351 and -0.557773351.Subproblem 1
a + -0.3333333334 = 0.557773351 Simplifying a + -0.3333333334 = 0.557773351 Reorder the terms: -0.3333333334 + a = 0.557773351 Solving -0.3333333334 + a = 0.557773351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = 0.557773351 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = 0.557773351 + 0.3333333334 a = 0.557773351 + 0.3333333334 Combine like terms: 0.557773351 + 0.3333333334 = 0.8911066844 a = 0.8911066844 Simplifying a = 0.8911066844Subproblem 2
a + -0.3333333334 = -0.557773351 Simplifying a + -0.3333333334 = -0.557773351 Reorder the terms: -0.3333333334 + a = -0.557773351 Solving -0.3333333334 + a = -0.557773351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = -0.557773351 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = -0.557773351 + 0.3333333334 a = -0.557773351 + 0.3333333334 Combine like terms: -0.557773351 + 0.3333333334 = -0.2244400176 a = -0.2244400176 Simplifying a = -0.2244400176Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.8911066844, -0.2244400176}
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