Solve (x+6)-3*x+2=(x+6)*x

$$( x + 6 ) - 3 \cdot x + 2 = ( x + 6 ) \cdot x$$

Solve for x

$$x=2\sqrt{6}-4\approx 0.898979486$$

$$x=-2\sqrt{6}-4\approx -8.898979486$$

Steps Using the Quadratic Formula

Combine x and -3x to get -2x.

$$-2x+6+2=\left(x+6\right)x$$

Add 6 and 2 to get 8.

$$-2x+8=\left(x+6\right)x$$

Use the distributive property to multiply x+6 by x.

$$-2x+8=x^{2}+6x$$

Subtract x^{2} from both sides.

$$-2x+8-x^{2}=6x$$

Subtract 6x from both sides.

$$-2x+8-x^{2}-6x=0$$

Combine -2x and -6x to get -8x.

$$-8x+8-x^{2}=0$$

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.

$$-x^{2}-8x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$x=\frac{-\left(-8\right)±\sqrt{64+32}}{2\left(-1\right)}$$

Add 64 to 32.

$$x=\frac{-\left(-8\right)±\sqrt{96}}{2\left(-1\right)}$$

Take the square root of 96.

$$x=\frac{-\left(-8\right)±4\sqrt{6}}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±4\sqrt{6}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±4\sqrt{6}}{-2}$$

Now solve the equation x=\frac{8±4\sqrt{6}}{-2} when ± is plus. Add 8 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}+8}{-2}$$

Divide 8+4\sqrt{6} by -2.

$$x=-2\sqrt{6}-4$$

Now solve the equation x=\frac{8±4\sqrt{6}}{-2} when ± is minus. Subtract 4\sqrt{6} from 8.

$$x=\frac{8-4\sqrt{6}}{-2}$$

Divide 8-4\sqrt{6} by -2.

$$x=2\sqrt{6}-4$$

The equation is now solved.

$$x=-2\sqrt{6}-4 x=2\sqrt{6}-4$$

Steps for Completing the Square

Combine x and -3x to get -2x.

$$-2x+6+2=\left(x+6\right)x$$

Add 6 and 2 to get 8.

$$-2x+8=\left(x+6\right)x$$

Use the distributive property to multiply x+6 by x.

$$-2x+8=x^{2}+6x$$

Subtract x^{2} from both sides.

$$-2x+8-x^{2}=6x$$

Subtract 6x from both sides.

$$-2x+8-x^{2}-6x=0$$

Combine -2x and -6x to get -8x.

$$-8x+8-x^{2}=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$-8x-x^{2}=-8$$

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.

$$-x^{2}-8x=-8$$

Divide both sides by -1.

$$\frac{-x^{2}-8x}{-1}=-\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=-\frac{8}{-1}$$

Divide -8 by -1.

$$x^{2}+8x=-\frac{8}{-1}$$

Divide -8 by -1.

$$x^{2}+8x=8$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=8+4^{2}$$

Square 4.

$$x^{2}+8x+16=8+16$$

Add 8 to 16.

$$x^{2}+8x+16=24$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=24$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{24}$$

Simplify.

$$x+4=2\sqrt{6} x+4=-2\sqrt{6}$$

Subtract 4 from both sides of the equation.

$$x=2\sqrt{6}-4 x=-2\sqrt{6}-4$$