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Simplifying 25x4 + -30x2 + 9 = (4x + 3)(4x + -3) + 9x2 + 3 Reorder the terms: 9 + -30x2 + 25x4 = (4x + 3)(4x + -3) + 9x2 + 3 Reorder the terms: 9 + -30x2 + 25x4 = (3 + 4x)(4x + -3) + 9x2 + 3 Reorder the terms: 9 + -30x2 + 25x4 = (3 + 4x)(-3 + 4x) + 9x2 + 3 Multiply (3 + 4x) * (-3 + 4x) 9 + -30x2 + 25x4 = (3(-3 + 4x) + 4x * (-3 + 4x)) + 9x2 + 3 9 + -30x2 + 25x4 = ((-3 * 3 + 4x * 3) + 4x * (-3 + 4x)) + 9x2 + 3 9 + -30x2 + 25x4 = ((-9 + 12x) + 4x * (-3 + 4x)) + 9x2 + 3 9 + -30x2 + 25x4 = (-9 + 12x + (-3 * 4x + 4x * 4x)) + 9x2 + 3 9 + -30x2 + 25x4 = (-9 + 12x + (-12x + 16x2)) + 9x2 + 3 Combine like terms: 12x + -12x = 0 9 + -30x2 + 25x4 = (-9 + 0 + 16x2) + 9x2 + 3 9 + -30x2 + 25x4 = (-9 + 16x2) + 9x2 + 3 Reorder the terms: 9 + -30x2 + 25x4 = -9 + 3 + 16x2 + 9x2 Combine like terms: -9 + 3 = -6 9 + -30x2 + 25x4 = -6 + 16x2 + 9x2 Combine like terms: 16x2 + 9x2 = 25x2 9 + -30x2 + 25x4 = -6 + 25x2 Solving 9 + -30x2 + 25x4 = -6 + 25x2 Solving for variable 'x'. Reorder the terms: 9 + 6 + -30x2 + -25x2 + 25x4 = -6 + 25x2 + 6 + -25x2 Combine like terms: 9 + 6 = 15 15 + -30x2 + -25x2 + 25x4 = -6 + 25x2 + 6 + -25x2 Combine like terms: -30x2 + -25x2 = -55x2 15 + -55x2 + 25x4 = -6 + 25x2 + 6 + -25x2 Reorder the terms: 15 + -55x2 + 25x4 = -6 + 6 + 25x2 + -25x2 Combine like terms: -6 + 6 = 0 15 + -55x2 + 25x4 = 0 + 25x2 + -25x2 15 + -55x2 + 25x4 = 25x2 + -25x2 Combine like terms: 25x2 + -25x2 = 0 15 + -55x2 + 25x4 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(3 + -11x2 + 5x4) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(3 + -11x2 + 5x4)' equal to zero and attempt to solve: Simplifying 3 + -11x2 + 5x4 = 0 Solving 3 + -11x2 + 5x4 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.6 + -2.2x2 + x4 = 0 Move the constant term to the right: Add '-0.6' to each side of the equation. 0.6 + -2.2x2 + -0.6 + x4 = 0 + -0.6 Reorder the terms: 0.6 + -0.6 + -2.2x2 + x4 = 0 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + -2.2x2 + x4 = 0 + -0.6 -2.2x2 + x4 = 0 + -0.6 Combine like terms: 0 + -0.6 = -0.6 -2.2x2 + x4 = -0.6 The x term is -2.2x2. Take half its coefficient (-1.1). Square it (1.21) and add it to both sides. Add '1.21' to each side of the equation. -2.2x2 + 1.21 + x4 = -0.6 + 1.21 Reorder the terms: 1.21 + -2.2x2 + x4 = -0.6 + 1.21 Combine like terms: -0.6 + 1.21 = 0.61 1.21 + -2.2x2 + x4 = 0.61 Factor a perfect square on the left side: (x2 + -1.1)(x2 + -1.1) = 0.61 Calculate the square root of the right side: 0.781024968 Break this problem into two subproblems by setting (x2 + -1.1) equal to 0.781024968 and -0.781024968.Subproblem 1
x2 + -1.1 = 0.781024968 Simplifying x2 + -1.1 = 0.781024968 Reorder the terms: -1.1 + x2 = 0.781024968 Solving -1.1 + x2 = 0.781024968 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.1' to each side of the equation. -1.1 + 1.1 + x2 = 0.781024968 + 1.1 Combine like terms: -1.1 + 1.1 = 0.0 0.0 + x2 = 0.781024968 + 1.1 x2 = 0.781024968 + 1.1 Combine like terms: 0.781024968 + 1.1 = 1.881024968 x2 = 1.881024968 Simplifying x2 = 1.881024968 Take the square root of each side: x = {-1.371504637, 1.371504637}Subproblem 2
x2 + -1.1 = -0.781024968 Simplifying x2 + -1.1 = -0.781024968 Reorder the terms: -1.1 + x2 = -0.781024968 Solving -1.1 + x2 = -0.781024968 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.1' to each side of the equation. -1.1 + 1.1 + x2 = -0.781024968 + 1.1 Combine like terms: -1.1 + 1.1 = 0.0 0.0 + x2 = -0.781024968 + 1.1 x2 = -0.781024968 + 1.1 Combine like terms: -0.781024968 + 1.1 = 0.318975032 x2 = 0.318975032 Simplifying x2 = 0.318975032 Take the square root of each side: x = {-0.564778746, 0.564778746}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.371504637, 1.371504637, -0.564778746, 0.564778746}Solution
x = {-1.371504637, 1.371504637, -0.564778746, 0.564778746}
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