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Simplifying -7a(a + 6) + 2(5 + 5a) = -53 Reorder the terms: -7a(6 + a) + 2(5 + 5a) = -53 (6 * -7a + a * -7a) + 2(5 + 5a) = -53 (-42a + -7a2) + 2(5 + 5a) = -53 -42a + -7a2 + (5 * 2 + 5a * 2) = -53 -42a + -7a2 + (10 + 10a) = -53 Reorder the terms: 10 + -42a + 10a + -7a2 = -53 Combine like terms: -42a + 10a = -32a 10 + -32a + -7a2 = -53 Solving 10 + -32a + -7a2 = -53 Solving for variable 'a'. Reorder the terms: 10 + 53 + -32a + -7a2 = -53 + 53 Combine like terms: 10 + 53 = 63 63 + -32a + -7a2 = -53 + 53 Combine like terms: -53 + 53 = 0 63 + -32a + -7a2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -9 + 4.571428571a + a2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 4.571428571a + 9 + a2 = 0 + 9 Reorder the terms: -9 + 9 + 4.571428571a + a2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 4.571428571a + a2 = 0 + 9 4.571428571a + a2 = 0 + 9 Combine like terms: 0 + 9 = 9 4.571428571a + a2 = 9 The a term is 4.571428571a. Take half its coefficient (2.285714286). Square it (5.224489797) and add it to both sides. Add '5.224489797' to each side of the equation. 4.571428571a + 5.224489797 + a2 = 9 + 5.224489797 Reorder the terms: 5.224489797 + 4.571428571a + a2 = 9 + 5.224489797 Combine like terms: 9 + 5.224489797 = 14.224489797 5.224489797 + 4.571428571a + a2 = 14.224489797 Factor a perfect square on the left side: (a + 2.285714286)(a + 2.285714286) = 14.224489797 Calculate the square root of the right side: 3.771536795 Break this problem into two subproblems by setting (a + 2.285714286) equal to 3.771536795 and -3.771536795.Subproblem 1
a + 2.285714286 = 3.771536795 Simplifying a + 2.285714286 = 3.771536795 Reorder the terms: 2.285714286 + a = 3.771536795 Solving 2.285714286 + a = 3.771536795 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = 3.771536795 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = 3.771536795 + -2.285714286 a = 3.771536795 + -2.285714286 Combine like terms: 3.771536795 + -2.285714286 = 1.485822509 a = 1.485822509 Simplifying a = 1.485822509Subproblem 2
a + 2.285714286 = -3.771536795 Simplifying a + 2.285714286 = -3.771536795 Reorder the terms: 2.285714286 + a = -3.771536795 Solving 2.285714286 + a = -3.771536795 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = -3.771536795 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = -3.771536795 + -2.285714286 a = -3.771536795 + -2.285714286 Combine like terms: -3.771536795 + -2.285714286 = -6.057251081 a = -6.057251081 Simplifying a = -6.057251081Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.485822509, -6.057251081}
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