Solve x^2+{left(31-3x/2right)}^2-8x-6(31-3x/2)+y=0

$${ x }^{ 2 } + { \left( \frac{ 31-3x }{ 2 } \right) }^{ 2 } -8x-6( \frac{ 31-3x }{ 2 } )+y=0$$

Solve for y

$$y=-\frac{13x^{2}}{4}+\frac{91x}{2}-\frac{589}{4}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2\left(x^{2}+\left(\frac{31-3x}{2}\right)^{2}-8x-6\times \frac{31-3x}{2}\right)+2y=0$$

To raise \frac{31-3x}{2} to a power, raise both numerator and denominator to the power and then divide.

$$2\left(x^{2}+\frac{\left(31-3x\right)^{2}}{2^{2}}-8x-6\times \frac{31-3x}{2}\right)+2y=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}-8x times \frac{2^{2}}{2^{2}}.

$$2\left(\frac{\left(x^{2}-8x\right)\times 2^{2}}{2^{2}}+\frac{\left(31-3x\right)^{2}}{2^{2}}-6\times \frac{31-3x}{2}\right)+2y=0$$

Since \frac{\left(x^{2}-8x\right)\times 2^{2}}{2^{2}} and \frac{\left(31-3x\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$2\left(\frac{\left(x^{2}-8x\right)\times 2^{2}+\left(31-3x\right)^{2}}{2^{2}}-6\times \frac{31-3x}{2}\right)+2y=0$$

Do the multiplications in \left(x^{2}-8x\right)\times 2^{2}+\left(31-3x\right)^{2}.

$$2\left(\frac{4x^{2}-32x+961-186x+9x^{2}}{2^{2}}-6\times \frac{31-3x}{2}\right)+2y=0$$

Combine like terms in 4x^{2}-32x+961-186x+9x^{2}.

$$2\left(\frac{13x^{2}-218x+961}{2^{2}}-6\times \frac{31-3x}{2}\right)+2y=0$$

Cancel out 2, the greatest common factor in 6 and 2.

$$2\left(\frac{13x^{2}-218x+961}{2^{2}}-3\left(31-3x\right)\right)+2y=0$$

Use the distributive property to multiply -3 by 31-3x.

$$2\left(\frac{13x^{2}-218x+961}{2^{2}}-93+9x\right)+2y=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -93+9x times \frac{2^{2}}{2^{2}}.

$$2\left(\frac{13x^{2}-218x+961}{2^{2}}+\frac{\left(-93+9x\right)\times 2^{2}}{2^{2}}\right)+2y=0$$

Since \frac{13x^{2}-218x+961}{2^{2}} and \frac{\left(-93+9x\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$2\times \frac{13x^{2}-218x+961+\left(-93+9x\right)\times 2^{2}}{2^{2}}+2y=0$$

Do the multiplications in 13x^{2}-218x+961+\left(-93+9x\right)\times 2^{2}.

$$2\times \frac{13x^{2}-218x+961-372+36x}{2^{2}}+2y=0$$

Combine like terms in 13x^{2}-218x+961-372+36x.

$$2\times \frac{13x^{2}-182x+589}{2^{2}}+2y=0$$

Express 2\times \frac{13x^{2}-182x+589}{2^{2}} as a single fraction.

$$\frac{2\left(13x^{2}-182x+589\right)}{2^{2}}+2y=0$$

Cancel out 2 in both numerator and denominator.

$$\frac{13x^{2}-182x+589}{2}+2y=0$$

Divide each term of 13x^{2}-182x+589 by 2 to get \frac{13}{2}x^{2}-91x+\frac{589}{2}.

$$\frac{13}{2}x^{2}-91x+\frac{589}{2}+2y=0$$

Subtract \frac{13}{2}x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-91x+\frac{589}{2}+2y=-\frac{13}{2}x^{2}$$

Add 91x to both sides.

$$\frac{589}{2}+2y=-\frac{13}{2}x^{2}+91x$$

Subtract \frac{589}{2} from both sides.

$$2y=-\frac{13}{2}x^{2}+91x-\frac{589}{2}$$

The equation is in standard form.

$$2y=-\frac{13x^{2}}{2}+91x-\frac{589}{2}$$

Divide both sides by 2.

$$\frac{2y}{2}=\frac{-\frac{13x^{2}}{2}+91x-\frac{589}{2}}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$y=\frac{-\frac{13x^{2}}{2}+91x-\frac{589}{2}}{2}$$

Divide -\frac{13x^{2}}{2}+91x-\frac{589}{2} by 2.

$$y=-\frac{13x^{2}}{4}+\frac{91x}{2}-\frac{589}{4}$$

Solve for x (complex solution)

$$x=\frac{2\sqrt{156-13y}}{13}+7$$

$$x=-\frac{2\sqrt{156-13y}}{13}+7$$

Solve for x

$$x=\frac{2\sqrt{156-13y}}{13}+7$$

$$x=-\frac{2\sqrt{156-13y}}{13}+7\text{, }y\leq 12$$