Solve y^2/2-2y+2=6y

$$\frac { y ^ { 2 } } { 2 } - 2 y + 2 = 6 y$$

Solve for y

$$y=2\sqrt{15}+8\approx 15.745966692$$

$$y=8-2\sqrt{15}\approx 0.254033308$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$y^{2}-4y+4=12y$$

Subtract 12y from both sides.

$$y^{2}-4y+4-12y=0$$

Combine -4y and -12y to get -16y.

$$y^{2}-16y+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4}}{2}$$

Square -16.

$$y=\frac{-\left(-16\right)±\sqrt{256-4\times 4}}{2}$$

Multiply -4 times 4.

$$y=\frac{-\left(-16\right)±\sqrt{256-16}}{2}$$

Add 256 to -16.

$$y=\frac{-\left(-16\right)±\sqrt{240}}{2}$$

Take the square root of 240.

$$y=\frac{-\left(-16\right)±4\sqrt{15}}{2}$$

The opposite of -16 is 16.

$$y=\frac{16±4\sqrt{15}}{2}$$

Now solve the equation y=\frac{16±4\sqrt{15}}{2} when ± is plus. Add 16 to 4\sqrt{15}.

$$y=\frac{4\sqrt{15}+16}{2}$$

Divide 16+4\sqrt{15} by 2.

$$y=2\sqrt{15}+8$$

Now solve the equation y=\frac{16±4\sqrt{15}}{2} when ± is minus. Subtract 4\sqrt{15} from 16.

$$y=\frac{16-4\sqrt{15}}{2}$$

Divide 16-4\sqrt{15} by 2.

$$y=8-2\sqrt{15}$$

The equation is now solved.

$$y=2\sqrt{15}+8 y=8-2\sqrt{15}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$y^{2}-4y+4=12y$$

Subtract 12y from both sides.

$$y^{2}-4y+4-12y=0$$

Combine -4y and -12y to get -16y.

$$y^{2}-16y+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}-16y=-4$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-16y+\left(-8\right)^{2}=-4+\left(-8\right)^{2}$$

Square -8.

$$y^{2}-16y+64=-4+64$$

Add -4 to 64.

$$y^{2}-16y+64=60$$

Factor y^{2}-16y+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-8\right)^{2}=60$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-8\right)^{2}}=\sqrt{60}$$

Simplify.

$$y-8=2\sqrt{15} y-8=-2\sqrt{15}$$

Add 8 to both sides of the equation.

$$y=2\sqrt{15}+8 y=8-2\sqrt{15}$$