Solve 3x=68+xx/2

$$3x=68+x \frac{ x }{ 2 }$$

Solve for x (complex solution)

$$x=3+\sqrt{127}i\approx 3+11.26942767i$$

$$x=-\sqrt{127}i+3\approx 3-11.26942767i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$6x=136+xx$$

Multiply x and x to get x^{2}.

$$6x=136+x^{2}$$

Subtract 136 from both sides.

$$6x-136=x^{2}$$

Subtract x^{2} from both sides.

$$6x-136-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+6x-136=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and -136 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\left(-136\right)}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\left(-136\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\left(-136\right)}}{2\left(-1\right)}$$

Multiply 4 times -136.

$$x=\frac{-6±\sqrt{36-544}}{2\left(-1\right)}$$

Add 36 to -544.

$$x=\frac{-6±\sqrt{-508}}{2\left(-1\right)}$$

Take the square root of -508.

$$x=\frac{-6±2\sqrt{127}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±2\sqrt{127}i}{-2}$$

Now solve the equation x=\frac{-6±2\sqrt{127}i}{-2} when ± is plus. Add -6 to 2i\sqrt{127}.

$$x=\frac{-6+2\sqrt{127}i}{-2}$$

Divide -6+2i\sqrt{127} by -2.

$$x=-\sqrt{127}i+3$$

Now solve the equation x=\frac{-6±2\sqrt{127}i}{-2} when ± is minus. Subtract 2i\sqrt{127} from -6.

$$x=\frac{-2\sqrt{127}i-6}{-2}$$

Divide -6-2i\sqrt{127} by -2.

$$x=3+\sqrt{127}i$$

The equation is now solved.

$$x=-\sqrt{127}i+3 x=3+\sqrt{127}i$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$6x=136+xx$$

Multiply x and x to get x^{2}.

$$6x=136+x^{2}$$

Subtract x^{2} from both sides.

$$6x-x^{2}=136$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+6x=136$$

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=\frac{136}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=\frac{136}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=\frac{136}{-1}$$

Divide 136 by -1.

$$x^{2}-6x=-136$$

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

$$x^{2}-6x+\left(-3\right)^{2}=-136+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-136+9$$

Add -136 to 9.

$$x^{2}-6x+9=-127$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=-127$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-127}$$

Simplify.

$$x-3=\sqrt{127}i x-3=-\sqrt{127}i$$

Add 3 to both sides of the equation.

$$x=3+\sqrt{127}i x=-\sqrt{127}i+3$$