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Simplifying 7a2 + 6a = 20 Reorder the terms: 6a + 7a2 = 20 Solving 6a + 7a2 = 20 Solving for variable 'a'. Reorder the terms: -20 + 6a + 7a2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + 6a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.857142857 + 0.8571428571a + a2 = 0 Move the constant term to the right: Add '2.857142857' to each side of the equation. -2.857142857 + 0.8571428571a + 2.857142857 + a2 = 0 + 2.857142857 Reorder the terms: -2.857142857 + 2.857142857 + 0.8571428571a + a2 = 0 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + 0.8571428571a + a2 = 0 + 2.857142857 0.8571428571a + a2 = 0 + 2.857142857 Combine like terms: 0 + 2.857142857 = 2.857142857 0.8571428571a + a2 = 2.857142857 The a term is 0.8571428571a. Take half its coefficient (0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. 0.8571428571a + 0.1836734694 + a2 = 2.857142857 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571a + a2 = 2.857142857 + 0.1836734694 Combine like terms: 2.857142857 + 0.1836734694 = 3.0408163264 0.1836734694 + 0.8571428571a + a2 = 3.0408163264 Factor a perfect square on the left side: (a + 0.4285714286)(a + 0.4285714286) = 3.0408163264 Calculate the square root of the right side: 1.743793659 Break this problem into two subproblems by setting (a + 0.4285714286) equal to 1.743793659 and -1.743793659.Subproblem 1
a + 0.4285714286 = 1.743793659 Simplifying a + 0.4285714286 = 1.743793659 Reorder the terms: 0.4285714286 + a = 1.743793659 Solving 0.4285714286 + a = 1.743793659 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + a = 1.743793659 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + a = 1.743793659 + -0.4285714286 a = 1.743793659 + -0.4285714286 Combine like terms: 1.743793659 + -0.4285714286 = 1.3152222304 a = 1.3152222304 Simplifying a = 1.3152222304Subproblem 2
a + 0.4285714286 = -1.743793659 Simplifying a + 0.4285714286 = -1.743793659 Reorder the terms: 0.4285714286 + a = -1.743793659 Solving 0.4285714286 + a = -1.743793659 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + a = -1.743793659 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + a = -1.743793659 + -0.4285714286 a = -1.743793659 + -0.4285714286 Combine like terms: -1.743793659 + -0.4285714286 = -2.1723650876 a = -2.1723650876 Simplifying a = -2.1723650876Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.3152222304, -2.1723650876}
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