4x^2+9x=292.5

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Solution for 4x^2+9x=292.5 equation:


Simplifying
4x2 + 9x = 292.5

Reorder the terms:
9x + 4x2 = 292.5

Solving
9x + 4x2 = 292.5

Solving for variable 'x'.

Reorder the terms:
-292.5 + 9x + 4x2 = 292.5 + -292.5

Combine like terms: 292.5 + -292.5 = 0.0
-292.5 + 9x + 4x2 = 0.0

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

Divide each side by '4'.
-73.125 + 2.25x + x2 = 0.0

Move the constant term to the right:

Add '73.125' to each side of the equation.
-73.125 + 2.25x + 73.125 + x2 = 0.0 + 73.125

Reorder the terms:
-73.125 + 73.125 + 2.25x + x2 = 0.0 + 73.125

Combine like terms: -73.125 + 73.125 = 0.000
0.000 + 2.25x + x2 = 0.0 + 73.125
2.25x + x2 = 0.0 + 73.125

Combine like terms: 0.0 + 73.125 = 73.125
2.25x + x2 = 73.125

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = 73.125 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = 73.125 + 1.265625

Combine like terms: 73.125 + 1.265625 = 74.390625
1.265625 + 2.25x + x2 = 74.390625

Factor a perfect square on the left side:
(x + 1.125)(x + 1.125) = 74.390625

Calculate the square root of the right side: 8.625

Break this problem into two subproblems by setting 
(x + 1.125) equal to 8.625 and -8.625.

Subproblem 1

x + 1.125 = 8.625 Simplifying x + 1.125 = 8.625 Reorder the terms: 1.125 + x = 8.625 Solving 1.125 + x = 8.625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = 8.625 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = 8.625 + -1.125 x = 8.625 + -1.125 Combine like terms: 8.625 + -1.125 = 7.5 x = 7.5 Simplifying x = 7.5

Subproblem 2

x + 1.125 = -8.625 Simplifying x + 1.125 = -8.625 Reorder the terms: 1.125 + x = -8.625 Solving 1.125 + x = -8.625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = -8.625 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = -8.625 + -1.125 x = -8.625 + -1.125 Combine like terms: -8.625 + -1.125 = -9.75 x = -9.75 Simplifying x = -9.75

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.5, -9.75}

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