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

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

Solve for x

$$x=\frac{4\sqrt{39}}{13}+7\approx 8.921537846$$

$$x=-\frac{4\sqrt{39}}{13}+7\approx 5.078462154$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

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

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

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

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

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.

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

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

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

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

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

Express 2\times \frac{13x^{2}-218x+961}{2^{2}} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

Use the distributive property to multiply -6 by 31-3x.

$$\frac{13x^{2}-218x+961}{2}-186+18x=0$$

Divide each term of 13x^{2}-218x+961 by 2 to get \frac{13}{2}x^{2}-109x+\frac{961}{2}.

$$\frac{13}{2}x^{2}-109x+\frac{961}{2}-186+18x=0$$

Subtract 186 from \frac{961}{2} to get \frac{589}{2}.

$$\frac{13}{2}x^{2}-109x+\frac{589}{2}+18x=0$$

Combine -109x and 18x to get -91x.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{13}{2} for a, -91 for b, and \frac{589}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-91\right)±\sqrt{\left(-91\right)^{2}-4\times \frac{13}{2}\times \frac{589}{2}}}{2\times \frac{13}{2}}$$

Square -91.

$$x=\frac{-\left(-91\right)±\sqrt{8281-4\times \frac{13}{2}\times \frac{589}{2}}}{2\times \frac{13}{2}}$$

Multiply -4 times \frac{13}{2}.

$$x=\frac{-\left(-91\right)±\sqrt{8281-26\times \frac{589}{2}}}{2\times \frac{13}{2}}$$

Multiply -26 times \frac{589}{2}.

$$x=\frac{-\left(-91\right)±\sqrt{8281-7657}}{2\times \frac{13}{2}}$$

Add 8281 to -7657.

$$x=\frac{-\left(-91\right)±\sqrt{624}}{2\times \frac{13}{2}}$$

Take the square root of 624.

$$x=\frac{-\left(-91\right)±4\sqrt{39}}{2\times \frac{13}{2}}$$

The opposite of -91 is 91.

$$x=\frac{91±4\sqrt{39}}{2\times \frac{13}{2}}$$

Multiply 2 times \frac{13}{2}.

$$x=\frac{91±4\sqrt{39}}{13}$$

Now solve the equation x=\frac{91±4\sqrt{39}}{13} when ± is plus. Add 91 to 4\sqrt{39}.

$$x=\frac{4\sqrt{39}+91}{13}$$

Divide 91+4\sqrt{39} by 13.

$$x=\frac{4\sqrt{39}}{13}+7$$

Now solve the equation x=\frac{91±4\sqrt{39}}{13} when ± is minus. Subtract 4\sqrt{39} from 91.

$$x=\frac{91-4\sqrt{39}}{13}$$

Divide 91-4\sqrt{39} by 13.

$$x=-\frac{4\sqrt{39}}{13}+7$$

The equation is now solved.

$$x=\frac{4\sqrt{39}}{13}+7 x=-\frac{4\sqrt{39}}{13}+7$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

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

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

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

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

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.

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

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

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

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

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

Express 2\times \frac{13x^{2}-218x+961}{2^{2}} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

Use the distributive property to multiply -6 by 31-3x.

$$\frac{13x^{2}-218x+961}{2}-186+18x=0$$

Divide each term of 13x^{2}-218x+961 by 2 to get \frac{13}{2}x^{2}-109x+\frac{961}{2}.

$$\frac{13}{2}x^{2}-109x+\frac{961}{2}-186+18x=0$$

Subtract 186 from \frac{961}{2} to get \frac{589}{2}.

$$\frac{13}{2}x^{2}-109x+\frac{589}{2}+18x=0$$

Combine -109x and 18x to get -91x.

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

Subtract \frac{589}{2} from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides of the equation by \frac{13}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

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

Dividing by \frac{13}{2} undoes the multiplication by \frac{13}{2}.

$$x^{2}+\left(-\frac{91}{\frac{13}{2}}\right)x=-\frac{\frac{589}{2}}{\frac{13}{2}}$$

Divide -91 by \frac{13}{2} by multiplying -91 by the reciprocal of \frac{13}{2}.

$$x^{2}-14x=-\frac{\frac{589}{2}}{\frac{13}{2}}$$

Divide -\frac{589}{2} by \frac{13}{2} by multiplying -\frac{589}{2} by the reciprocal of \frac{13}{2}.

$$x^{2}-14x=-\frac{589}{13}$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-\frac{589}{13}+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-\frac{589}{13}+49$$

Add -\frac{589}{13} to 49.

$$x^{2}-14x+49=\frac{48}{13}$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=\frac{48}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{\frac{48}{13}}$$

Simplify.

$$x-7=\frac{4\sqrt{39}}{13} x-7=-\frac{4\sqrt{39}}{13}$$

Add 7 to both sides of the equation.

$$x=\frac{4\sqrt{39}}{13}+7 x=-\frac{4\sqrt{39}}{13}+7$$