Solve x^2-4x+8+x^2/2-14x+100=-x/2-2x+52

$${ x }^{ 2 } -4x+8+ \frac{ { x }^{ 2 } }{ 2 } -14x+100 = - \frac{ x }{ 2 } -2x+52$$

Solve for x (complex solution)

$$x=\frac{31+\sqrt{383}i}{6}\approx 5.166666667+3.261730965i$$

$$x=\frac{-\sqrt{383}i+31}{6}\approx 5.166666667-3.261730965i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2x^{2}-8x+16+x^{2}-28x+200=-x-4x+104$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}-8x+16-28x+200=-x-4x+104$$

Combine -8x and -28x to get -36x.

$$3x^{2}-36x+16+200=-x-4x+104$$

Add 16 and 200 to get 216.

$$3x^{2}-36x+216=-x-4x+104$$

Add x to both sides.

$$3x^{2}-36x+216+x=-4x+104$$

Combine -36x and x to get -35x.

$$3x^{2}-35x+216=-4x+104$$

Add 4x to both sides.

$$3x^{2}-35x+216+4x=104$$

Combine -35x and 4x to get -31x.

$$3x^{2}-31x+216=104$$

Subtract 104 from both sides.

$$3x^{2}-31x+216-104=0$$

Subtract 104 from 216 to get 112.

$$3x^{2}-31x+112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -31 for b, and 112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 3\times 112}}{2\times 3}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 3\times 112}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-31\right)±\sqrt{961-12\times 112}}{2\times 3}$$

Multiply -12 times 112.

$$x=\frac{-\left(-31\right)±\sqrt{961-1344}}{2\times 3}$$

Add 961 to -1344.

$$x=\frac{-\left(-31\right)±\sqrt{-383}}{2\times 3}$$

Take the square root of -383.

$$x=\frac{-\left(-31\right)±\sqrt{383}i}{2\times 3}$$

The opposite of -31 is 31.

$$x=\frac{31±\sqrt{383}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{31±\sqrt{383}i}{6}$$

Now solve the equation x=\frac{31±\sqrt{383}i}{6} when ± is plus. Add 31 to i\sqrt{383}.

$$x=\frac{31+\sqrt{383}i}{6}$$

Now solve the equation x=\frac{31±\sqrt{383}i}{6} when ± is minus. Subtract i\sqrt{383} from 31.

$$x=\frac{-\sqrt{383}i+31}{6}$$

The equation is now solved.

$$x=\frac{31+\sqrt{383}i}{6} x=\frac{-\sqrt{383}i+31}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2x^{2}-8x+16+x^{2}-28x+200=-x-4x+104$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}-8x+16-28x+200=-x-4x+104$$

Combine -8x and -28x to get -36x.

$$3x^{2}-36x+16+200=-x-4x+104$$

Add 16 and 200 to get 216.

$$3x^{2}-36x+216=-x-4x+104$$

Add x to both sides.

$$3x^{2}-36x+216+x=-4x+104$$

Combine -36x and x to get -35x.

$$3x^{2}-35x+216=-4x+104$$

Add 4x to both sides.

$$3x^{2}-35x+216+4x=104$$

Combine -35x and 4x to get -31x.

$$3x^{2}-31x+216=104$$

Subtract 216 from both sides.

$$3x^{2}-31x=104-216$$

Subtract 216 from 104 to get -112.

$$3x^{2}-31x=-112$$

Divide both sides by 3.

$$\frac{3x^{2}-31x}{3}=-\frac{112}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{31}{3}x=-\frac{112}{3}$$

Divide -\frac{31}{3}, the coefficient of the x term, by 2 to get -\frac{31}{6}. Then add the square of -\frac{31}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{3}x+\left(-\frac{31}{6}\right)^{2}=-\frac{112}{3}+\left(-\frac{31}{6}\right)^{2}$$

Square -\frac{31}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=-\frac{112}{3}+\frac{961}{36}$$

Add -\frac{112}{3} to \frac{961}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=-\frac{383}{36}$$

Factor x^{2}-\frac{31}{3}x+\frac{961}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{6}\right)^{2}=-\frac{383}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{6}\right)^{2}}=\sqrt{-\frac{383}{36}}$$

Simplify.

$$x-\frac{31}{6}=\frac{\sqrt{383}i}{6} x-\frac{31}{6}=-\frac{\sqrt{383}i}{6}$$

Add \frac{31}{6} to both sides of the equation.

$$x=\frac{31+\sqrt{383}i}{6} x=\frac{-\sqrt{383}i+31}{6}$$