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Simplifying 0 = 4x2 + 7x + 1 Reorder the terms: 0 = 1 + 7x + 4x2 Solving 0 = 1 + 7x + 4x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + -7x + -4x2 = 1 + 7x + 4x2 + -1 + -7x + -4x2 Reorder the terms: -1 + -7x + -4x2 = 1 + -1 + 7x + -7x + 4x2 + -4x2 Combine like terms: 1 + -1 = 0 -1 + -7x + -4x2 = 0 + 7x + -7x + 4x2 + -4x2 -1 + -7x + -4x2 = 7x + -7x + 4x2 + -4x2 Combine like terms: 7x + -7x = 0 -1 + -7x + -4x2 = 0 + 4x2 + -4x2 -1 + -7x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 -1 + -7x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 7x + 4x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 7x + 4x2)' equal to zero and attempt to solve: Simplifying 1 + 7x + 4x2 = 0 Solving 1 + 7x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + 1.75x + x2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 1.75x + -0.25 + x2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 1.75x + x2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 1.75x + x2 = 0 + -0.25 1.75x + x2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 1.75x + x2 = -0.25 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = -0.25 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = -0.25 + 0.765625 Combine like terms: -0.25 + 0.765625 = 0.515625 0.765625 + 1.75x + x2 = 0.515625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (x + 0.875) equal to 0.718070331 and -0.718070331.Subproblem 1
x + 0.875 = 0.718070331 Simplifying x + 0.875 = 0.718070331 Reorder the terms: 0.875 + x = 0.718070331 Solving 0.875 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 0.718070331 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 0.718070331 + -0.875 x = 0.718070331 + -0.875 Combine like terms: 0.718070331 + -0.875 = -0.156929669 x = -0.156929669 Simplifying x = -0.156929669Subproblem 2
x + 0.875 = -0.718070331 Simplifying x + 0.875 = -0.718070331 Reorder the terms: 0.875 + x = -0.718070331 Solving 0.875 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -0.718070331 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -0.718070331 + -0.875 x = -0.718070331 + -0.875 Combine like terms: -0.718070331 + -0.875 = -1.593070331 x = -1.593070331 Simplifying x = -1.593070331Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.156929669, -1.593070331}Solution
x = {-0.156929669, -1.593070331}
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