24x+2x^2=x^2+16

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Solution for 24x+2x^2=x^2+16 equation:


Simplifying
24x + 2x2 = x2 + 16

Reorder the terms:
24x + 2x2 = 16 + x2

Solving
24x + 2x2 = 16 + x2

Solving for variable 'x'.

Reorder the terms:
-16 + 24x + 2x2 + -1x2 = 16 + x2 + -16 + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
-16 + 24x + 1x2 = 16 + x2 + -16 + -1x2

Reorder the terms:
-16 + 24x + 1x2 = 16 + -16 + x2 + -1x2

Combine like terms: 16 + -16 = 0
-16 + 24x + 1x2 = 0 + x2 + -1x2
-16 + 24x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-16 + 24x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 24x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + 24x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 24x + x2 = 0 + 16
24x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
24x + x2 = 16

The x term is 24x.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24x + 144 + x2 = 16 + 144

Reorder the terms:
144 + 24x + x2 = 16 + 144

Combine like terms: 16 + 144 = 160
144 + 24x + x2 = 160

Factor a perfect square on the left side:
(x + 12)(x + 12) = 160

Calculate the square root of the right side: 12.649110641

Break this problem into two subproblems by setting 
(x + 12) equal to 12.649110641 and -12.649110641.

Subproblem 1

x + 12 = 12.649110641 Simplifying x + 12 = 12.649110641 Reorder the terms: 12 + x = 12.649110641 Solving 12 + x = 12.649110641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = 12.649110641 + -12 Combine like terms: 12 + -12 = 0 0 + x = 12.649110641 + -12 x = 12.649110641 + -12 Combine like terms: 12.649110641 + -12 = 0.649110641 x = 0.649110641 Simplifying x = 0.649110641

Subproblem 2

x + 12 = -12.649110641 Simplifying x + 12 = -12.649110641 Reorder the terms: 12 + x = -12.649110641 Solving 12 + x = -12.649110641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = -12.649110641 + -12 Combine like terms: 12 + -12 = 0 0 + x = -12.649110641 + -12 x = -12.649110641 + -12 Combine like terms: -12.649110641 + -12 = -24.649110641 x = -24.649110641 Simplifying x = -24.649110641

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.649110641, -24.649110641}

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