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Simplifying 6.75(0.12 + -0.70x + x2) = x (0.12 * 6.75 + -0.70x * 6.75 + x2 * 6.75) = x (0.81 + -4.725x + 6.75x2) = x Solving 0.81 + -4.725x + 6.75x2 = x Solving for variable 'x'. Reorder the terms: 0.81 + -4.725x + -1x + 6.75x2 = x + -1x Combine like terms: -4.725x + -1x = -5.725x 0.81 + -5.725x + 6.75x2 = x + -1x Combine like terms: x + -1x = 0 0.81 + -5.725x + 6.75x2 = 0 Begin completing the square. Divide all terms by 6.75 the coefficient of the squared term: Divide each side by '6.75'. 0.12 + -0.8481481481x + x2 = 0 Move the constant term to the right: Add '-0.12' to each side of the equation. 0.12 + -0.8481481481x + -0.12 + x2 = 0 + -0.12 Reorder the terms: 0.12 + -0.12 + -0.8481481481x + x2 = 0 + -0.12 Combine like terms: 0.12 + -0.12 = 0.00 0.00 + -0.8481481481x + x2 = 0 + -0.12 -0.8481481481x + x2 = 0 + -0.12 Combine like terms: 0 + -0.12 = -0.12 -0.8481481481x + x2 = -0.12 The x term is -0.8481481481x. Take half its coefficient (-0.4240740741). Square it (0.1798388203) and add it to both sides. Add '0.1798388203' to each side of the equation. -0.8481481481x + 0.1798388203 + x2 = -0.12 + 0.1798388203 Reorder the terms: 0.1798388203 + -0.8481481481x + x2 = -0.12 + 0.1798388203 Combine like terms: -0.12 + 0.1798388203 = 0.0598388203 0.1798388203 + -0.8481481481x + x2 = 0.0598388203 Factor a perfect square on the left side: (x + -0.4240740741)(x + -0.4240740741) = 0.0598388203 Calculate the square root of the right side: 0.244619746 Break this problem into two subproblems by setting (x + -0.4240740741) equal to 0.244619746 and -0.244619746.Subproblem 1
x + -0.4240740741 = 0.244619746 Simplifying x + -0.4240740741 = 0.244619746 Reorder the terms: -0.4240740741 + x = 0.244619746 Solving -0.4240740741 + x = 0.244619746 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4240740741' to each side of the equation. -0.4240740741 + 0.4240740741 + x = 0.244619746 + 0.4240740741 Combine like terms: -0.4240740741 + 0.4240740741 = 0.0000000000 0.0000000000 + x = 0.244619746 + 0.4240740741 x = 0.244619746 + 0.4240740741 Combine like terms: 0.244619746 + 0.4240740741 = 0.6686938201 x = 0.6686938201 Simplifying x = 0.6686938201Subproblem 2
x + -0.4240740741 = -0.244619746 Simplifying x + -0.4240740741 = -0.244619746 Reorder the terms: -0.4240740741 + x = -0.244619746 Solving -0.4240740741 + x = -0.244619746 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4240740741' to each side of the equation. -0.4240740741 + 0.4240740741 + x = -0.244619746 + 0.4240740741 Combine like terms: -0.4240740741 + 0.4240740741 = 0.0000000000 0.0000000000 + x = -0.244619746 + 0.4240740741 x = -0.244619746 + 0.4240740741 Combine like terms: -0.244619746 + 0.4240740741 = 0.1794543281 x = 0.1794543281 Simplifying x = 0.1794543281Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6686938201, 0.1794543281}
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