8x^2=9x+16

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Solution for 8x^2=9x+16 equation:


Simplifying
8x2 = 9x + 16

Reorder the terms:
8x2 = 16 + 9x

Solving
8x2 = 16 + 9x

Solving for variable 'x'.

Reorder the terms:
-16 + -9x + 8x2 = 16 + 9x + -16 + -9x

Reorder the terms:
-16 + -9x + 8x2 = 16 + -16 + 9x + -9x

Combine like terms: 16 + -16 = 0
-16 + -9x + 8x2 = 0 + 9x + -9x
-16 + -9x + 8x2 = 9x + -9x

Combine like terms: 9x + -9x = 0
-16 + -9x + 8x2 = 0

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

Divide each side by '8'.
-2 + -1.125x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.125x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.125x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.125x + x2 = 0 + 2
-1.125x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.125x + x2 = 2

The x term is -1.125x.  Take half its coefficient (-0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
-1.125x + 0.31640625 + x2 = 2 + 0.31640625

Reorder the terms:
0.31640625 + -1.125x + x2 = 2 + 0.31640625

Combine like terms: 2 + 0.31640625 = 2.31640625
0.31640625 + -1.125x + x2 = 2.31640625

Factor a perfect square on the left side:
(x + -0.5625)(x + -0.5625) = 2.31640625

Calculate the square root of the right side: 1.521974458

Break this problem into two subproblems by setting 
(x + -0.5625) equal to 1.521974458 and -1.521974458.

Subproblem 1

x + -0.5625 = 1.521974458 Simplifying x + -0.5625 = 1.521974458 Reorder the terms: -0.5625 + x = 1.521974458 Solving -0.5625 + x = 1.521974458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = 1.521974458 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = 1.521974458 + 0.5625 x = 1.521974458 + 0.5625 Combine like terms: 1.521974458 + 0.5625 = 2.084474458 x = 2.084474458 Simplifying x = 2.084474458

Subproblem 2

x + -0.5625 = -1.521974458 Simplifying x + -0.5625 = -1.521974458 Reorder the terms: -0.5625 + x = -1.521974458 Solving -0.5625 + x = -1.521974458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = -1.521974458 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = -1.521974458 + 0.5625 x = -1.521974458 + 0.5625 Combine like terms: -1.521974458 + 0.5625 = -0.959474458 x = -0.959474458 Simplifying x = -0.959474458

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.084474458, -0.959474458}

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