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-2x^{2}-8x+9=y
Swap sides so that all variable terms are on the left hand side.
-2x^{2}-8x+9-y=0
Subtract y from both sides.
x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-2\right)\left(9-y\right)}}{2\left(-2\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -8 for b, and 9-y for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\left(-2\right)\left(9-y\right)}}{2\left(-2\right)}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64+8\left(9-y\right)}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-\left(-8\right)±\sqrt{64+72-8y}}{2\left(-2\right)}
Multiply 8 times 9-y.
x=\frac{-\left(-8\right)±\sqrt{136-8y}}{2\left(-2\right)}
Add 64 to 72-8y.
x=\frac{-\left(-8\right)±2\sqrt{34-2y}}{2\left(-2\right)}
Take the square root of 136-8y.
x=\frac{8±2\sqrt{34-2y}}{2\left(-2\right)}
The opposite of -8 is 8.
x=\frac{8±2\sqrt{34-2y}}{-4}
Multiply 2 times -2.
x=\frac{2\sqrt{34-2y}+8}{-4}
Now solve the equation x=\frac{8±2\sqrt{34-2y}}{-4} when ± is plus. Add 8 to 2\sqrt{34-2y}.
x=-\frac{\sqrt{34-2y}}{2}-2
Divide 8+2\sqrt{34-2y} by -4.
x=\frac{-2\sqrt{34-2y}+8}{-4}
Now solve the equation x=\frac{8±2\sqrt{34-2y}}{-4} when ± is minus. Subtract 2\sqrt{34-2y} from 8.
x=\frac{\sqrt{34-2y}}{2}-2
Divide 8-2\sqrt{34-2y} by -4.
x=-\frac{\sqrt{34-2y}}{2}-2 x=\frac{\sqrt{34-2y}}{2}-2
The equation is now solved.
-2x^{2}-8x+9=y
Swap sides so that all variable terms are on the left hand side.
-2x^{2}-8x=y-9
Subtract 9 from both sides.
\frac{-2x^{2}-8x}{-2}=\frac{y-9}{-2}
Divide both sides by -2.
x^{2}+\frac{-8}{-2}x=\frac{y-9}{-2}
Dividing by -2 undoes the multiplication by -2.
x^{2}+4x=\frac{y-9}{-2}
Divide -8 by -2.
x^{2}+4x=\frac{9-y}{2}
Divide y-9 by -2.
x^{2}+4x+2^{2}=\frac{9-y}{2}+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=\frac{9-y}{2}+4
Square 2.
x^{2}+4x+4=\frac{17-y}{2}
Add \frac{-y+9}{2} to 4.
\left(x+2\right)^{2}=\frac{17-y}{2}
Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+2\right)^{2}}=\sqrt{\frac{17-y}{2}}
Take the square root of both sides of the equation.
x+2=\frac{\sqrt{34-2y}}{2} x+2=-\frac{\sqrt{34-2y}}{2}
Simplify.
x=\frac{\sqrt{34-2y}}{2}-2 x=-\frac{\sqrt{34-2y}}{2}-2
Subtract 2 from both sides of the equation.