24.3(0.024+0.32x+x^2)=0.90

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Solution for 24.3(0.024+0.32x+x^2)=0.90 equation:


Simplifying
24.3(0.024 + 0.32x + x2) = 0.90
(0.024 * 24.3 + 0.32x * 24.3 + x2 * 24.3) = 0.90
(0.5832 + 7.776x + 24.3x2) = 0.90

Solving
0.5832 + 7.776x + 24.3x2 = 0.90

Solving for variable 'x'.

Reorder the terms:
0.5832 + -0.90 + 7.776x + 24.3x2 = 0.90 + -0.90

Combine like terms: 0.5832 + -0.90 = -0.3168
-0.3168 + 7.776x + 24.3x2 = 0.90 + -0.90

Combine like terms: 0.90 + -0.90 = 0.00
-0.3168 + 7.776x + 24.3x2 = 0.00

Begin completing the square.  Divide all terms by
24.3 the coefficient of the squared term: 

Divide each side by '24.3'.
-0.01303703704 + 0.32x + x2 = 0.0

Move the constant term to the right:

Add '0.01303703704' to each side of the equation.
-0.01303703704 + 0.32x + 0.01303703704 + x2 = 0.0 + 0.01303703704

Reorder the terms:
-0.01303703704 + 0.01303703704 + 0.32x + x2 = 0.0 + 0.01303703704

Combine like terms: -0.01303703704 + 0.01303703704 = 0.00000000000
0.00000000000 + 0.32x + x2 = 0.0 + 0.01303703704
0.32x + x2 = 0.0 + 0.01303703704

Combine like terms: 0.0 + 0.01303703704 = 0.01303703704
0.32x + x2 = 0.01303703704

The x term is 0.32x.  Take half its coefficient (0.16).
Square it (0.0256) and add it to both sides.

Add '0.0256' to each side of the equation.
0.32x + 0.0256 + x2 = 0.01303703704 + 0.0256

Reorder the terms:
0.0256 + 0.32x + x2 = 0.01303703704 + 0.0256

Combine like terms: 0.01303703704 + 0.0256 = 0.03863703704
0.0256 + 0.32x + x2 = 0.03863703704

Factor a perfect square on the left side:
(x + 0.16)(x + 0.16) = 0.03863703704

Calculate the square root of the right side: 0.196563061

Break this problem into two subproblems by setting 
(x + 0.16) equal to 0.196563061 and -0.196563061.

Subproblem 1

x + 0.16 = 0.196563061 Simplifying x + 0.16 = 0.196563061 Reorder the terms: 0.16 + x = 0.196563061 Solving 0.16 + x = 0.196563061 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.16' to each side of the equation. 0.16 + -0.16 + x = 0.196563061 + -0.16 Combine like terms: 0.16 + -0.16 = 0.00 0.00 + x = 0.196563061 + -0.16 x = 0.196563061 + -0.16 Combine like terms: 0.196563061 + -0.16 = 0.036563061 x = 0.036563061 Simplifying x = 0.036563061

Subproblem 2

x + 0.16 = -0.196563061 Simplifying x + 0.16 = -0.196563061 Reorder the terms: 0.16 + x = -0.196563061 Solving 0.16 + x = -0.196563061 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.16' to each side of the equation. 0.16 + -0.16 + x = -0.196563061 + -0.16 Combine like terms: 0.16 + -0.16 = 0.00 0.00 + x = -0.196563061 + -0.16 x = -0.196563061 + -0.16 Combine like terms: -0.196563061 + -0.16 = -0.356563061 x = -0.356563061 Simplifying x = -0.356563061

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.036563061, -0.356563061}

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