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Simplifying (4a * 4a + -3b * 3b)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: (4 * 4a * a + -3b * 3b)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply 4 * 4 (16a * a + -3b * 3b)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply a * a (16a2 + -3b * 3b)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: (16a2 + -3 * 3b * b)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply -3 * 3 (16a2 + -9b * b)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply b * b (16a2 + -9b2)(4a * 4a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: (16a2 + -9b2)(4 * 4a * a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply 4 * 4 (16a2 + -9b2)(16a * a + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply a * a (16a2 + -9b2)(16a2 + -3b * 3b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: (16a2 + -9b2)(16a2 + -3 * 3b * b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply -3 * 3 (16a2 + -9b2)(16a2 + -9b * b) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply b * b (16a2 + -9b2)(16a2 + -9b2) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply (16a2 + -9b2) * (16a2 + -9b2) (16a2 * (16a2 + -9b2) + -9b2 * (16a2 + -9b2)) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 ((16a2 * 16a2 + -9b2 * 16a2) + -9b2 * (16a2 + -9b2)) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms: ((-144a2b2 + 256a4) + -9b2 * (16a2 + -9b2)) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 ((-144a2b2 + 256a4) + -9b2 * (16a2 + -9b2)) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 (-144a2b2 + 256a4 + (16a2 * -9b2 + -9b2 * -9b2)) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 (-144a2b2 + 256a4 + (-144a2b2 + 81b4)) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms: (-144a2b2 + -144a2b2 + 256a4 + 81b4) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Combine like terms: -144a2b2 + -144a2b2 = -288a2b2 (-288a2b2 + 256a4 + 81b4) + (3b * 3b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: -288a2b2 + 256a4 + 81b4 + (3 * 3b * b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply 3 * 3 -288a2b2 + 256a4 + 81b4 + (9b * b + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply b * b -288a2b2 + 256a4 + 81b4 + (9b2 + -4a * 4a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: -288a2b2 + 256a4 + 81b4 + (9b2 + -4 * 4a * a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply -4 * 4 -288a2b2 + 256a4 + 81b4 + (9b2 + -16a * a)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply a * a -288a2b2 + 256a4 + 81b4 + (9b2 + -16a2)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms: -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(3b * 3b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(3 * 3b * b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply 3 * 3 -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(9b * b + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Multiply b * b -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(9b2 + -4a * 4a) = 32a4 + -48a2b2 + 18b4 Reorder the terms for easier multiplication: -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(9b2 + -4 * 4a * a) = 32a4 + -48a2b2 + 18b4 Multiply -4 * 4 -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(9b2 + -16a * a) = 32a4 + -48a2b2 + 18b4 Multiply a * a -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(9b2 + -16a2) = 32a4 + -48a2b2 + 18b4 Reorder the terms: -288a2b2 + 256a4 + 81b4 + (-16a2 + 9b2)(-16a2 + 9b2) = 32a4 + -48a2b2 + 18b4 Multiply (-16a2 + 9b2) * (-16a2 + 9b2) -288a2b2 + 256a4 + 81b4 + (-16a2 * (-16a2 + 9b2) + 9b2 * (-16a2 + 9b2)) = 32a4 + -48a2b2 + 18b4 -288a2b2 + 256a4 + 81b4 + ((-16a2 * -16a2 + 9b2 * -16a2) + 9b2 * (-16a2 + 9b2)) = 32a4 + -48a2b2 + 18b4 Reorder the terms: -288a2b2 + 256a4 + 81b4 + ((-144a2b2 + 256a4) + 9b2 * (-16a2 + 9b2)) = 32a4 + -48a2b2 + 18b4 -288a2b2 + 256a4 + 81b4 + ((-144a2b2 + 256a4) + 9b2 * (-16a2 + 9b2)) = 32a4 + -48a2b2 + 18b4 -288a2b2 + 256a4 + 81b4 + (-144a2b2 + 256a4 + (-16a2 * 9b2 + 9b2 * 9b2)) = 32a4 + -48a2b2 + 18b4 -288a2b2 + 256a4 + 81b4 + (-144a2b2 + 256a4 + (-144a2b2 + 81b4)) = 32a4 + -48a2b2 + 18b4 Reorder the terms: -288a2b2 + 256a4 + 81b4 + (-144a2b2 + -144a2b2 + 256a4 + 81b4) = 32a4 + -48a2b2 + 18b4 Combine like terms: -144a2b2 + -144a2b2 = -288a2b2 -288a2b2 + 256a4 + 81b4 + (-288a2b2 + 256a4 + 81b4) = 32a4 + -48a2b2 + 18b4 Reorder the terms: -288a2b2 + -288a2b2 + 256a4 + 256a4 + 81b4 + 81b4 = 32a4 + -48a2b2 + 18b4 Combine like terms: -288a2b2 + -288a2b2 = -576a2b2 -576a2b2 + 256a4 + 256a4 + 81b4 + 81b4 = 32a4 + -48a2b2 + 18b4 Combine like terms: 256a4 + 256a4 = 512a4 -576a2b2 + 512a4 + 81b4 + 81b4 = 32a4 + -48a2b2 + 18b4 Combine like terms: 81b4 + 81b4 = 162b4 -576a2b2 + 512a4 + 162b4 = 32a4 + -48a2b2 + 18b4 Reorder the terms: -576a2b2 + 512a4 + 162b4 = -48a2b2 + 32a4 + 18b4 Solving -576a2b2 + 512a4 + 162b4 = -48a2b2 + 32a4 + 18b4 Solving for variable 'a'. Reorder the terms: -576a2b2 + 48a2b2 + 512a4 + -32a4 + 162b4 + -18b4 = -48a2b2 + 32a4 + 18b4 + 48a2b2 + -32a4 + -18b4 Combine like terms: -576a2b2 + 48a2b2 = -528a2b2 -528a2b2 + 512a4 + -32a4 + 162b4 + -18b4 = -48a2b2 + 32a4 + 18b4 + 48a2b2 + -32a4 + -18b4 Combine like terms: 512a4 + -32a4 = 480a4 -528a2b2 + 480a4 + 162b4 + -18b4 = -48a2b2 + 32a4 + 18b4 + 48a2b2 + -32a4 + -18b4 Combine like terms: 162b4 + -18b4 = 144b4 -528a2b2 + 480a4 + 144b4 = -48a2b2 + 32a4 + 18b4 + 48a2b2 + -32a4 + -18b4 Reorder the terms: -528a2b2 + 480a4 + 144b4 = -48a2b2 + 48a2b2 + 32a4 + -32a4 + 18b4 + -18b4 Combine like terms: -48a2b2 + 48a2b2 = 0 -528a2b2 + 480a4 + 144b4 = 0 + 32a4 + -32a4 + 18b4 + -18b4 -528a2b2 + 480a4 + 144b4 = 32a4 + -32a4 + 18b4 + -18b4 Combine like terms: 32a4 + -32a4 = 0 -528a2b2 + 480a4 + 144b4 = 0 + 18b4 + -18b4 -528a2b2 + 480a4 + 144b4 = 18b4 + -18b4 Combine like terms: 18b4 + -18b4 = 0 -528a2b2 + 480a4 + 144b4 = 0 Factor out the Greatest Common Factor (GCF), '48'. 48(-11a2b2 + 10a4 + 3b4) = 0 Factor a trinomial. 48((5a2 + -3b2)(2a2 + -1b2)) = 0 Ignore the factor 48.Subproblem 1
Set the factor '(5a2 + -3b2)' equal to zero and attempt to solve: Simplifying 5a2 + -3b2 = 0 Solving 5a2 + -3b2 = 0 Move all terms containing a to the left, all other terms to the right. Add '3b2' to each side of the equation. 5a2 + -3b2 + 3b2 = 0 + 3b2 Combine like terms: -3b2 + 3b2 = 0 5a2 + 0 = 0 + 3b2 5a2 = 0 + 3b2 Remove the zero: 5a2 = 3b2 Divide each side by '5'. a2 = 0.6b2 Simplifying a2 = 0.6b2 Take the square root of each side: a = {-0.774596669b, 0.774596669b}Subproblem 2
Set the factor '(2a2 + -1b2)' equal to zero and attempt to solve: Simplifying 2a2 + -1b2 = 0 Solving 2a2 + -1b2 = 0 Move all terms containing a to the left, all other terms to the right. Add 'b2' to each side of the equation. 2a2 + -1b2 + b2 = 0 + b2 Combine like terms: -1b2 + b2 = 0 2a2 + 0 = 0 + b2 2a2 = 0 + b2 Remove the zero: 2a2 = b2 Divide each side by '2'. a2 = 0.5b2 Simplifying a2 = 0.5b2 Take the square root of each side: a = {-0.707106781b, 0.707106781b}Solution
a = {-0.774596669b, 0.774596669b, -0.707106781b, 0.707106781b}
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