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-y^{2}+3y+5=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.
y=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-3±\sqrt{9-4\left(-1\right)\times 5}}{2\left(-1\right)}
Square 3.
y=\frac{-3±\sqrt{9+4\times 5}}{2\left(-1\right)}
Multiply -4 times -1.
y=\frac{-3±\sqrt{9+20}}{2\left(-1\right)}
Multiply 4 times 5.
y=\frac{-3±\sqrt{29}}{2\left(-1\right)}
Add 9 to 20.
y=\frac{-3±\sqrt{29}}{-2}
Multiply 2 times -1.
y=\frac{\sqrt{29}-3}{-2}
Now solve the equation y=\frac{-3±\sqrt{29}}{-2} when ± is plus. Add -3 to \sqrt{29}.
y=\frac{3-\sqrt{29}}{2}
Divide -3+\sqrt{29} by -2.
y=\frac{-\sqrt{29}-3}{-2}
Now solve the equation y=\frac{-3±\sqrt{29}}{-2} when ± is minus. Subtract \sqrt{29} from -3.
y=\frac{\sqrt{29}+3}{2}
Divide -3-\sqrt{29} by -2.
y=\frac{3-\sqrt{29}}{2} y=\frac{\sqrt{29}+3}{2}
The equation is now solved.
-y^{2}+3y+5=0
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.
-y^{2}+3y+5-5=-5
Subtract 5 from both sides of the equation.
-y^{2}+3y=-5
Subtracting 5 from itself leaves 0.
\frac{-y^{2}+3y}{-1}=-\frac{5}{-1}
Divide both sides by -1.
y^{2}+\frac{3}{-1}y=-\frac{5}{-1}
Dividing by -1 undoes the multiplication by -1.
y^{2}-3y=-\frac{5}{-1}
Divide 3 by -1.
y^{2}-3y=5
Divide -5 by -1.
y^{2}-3y+\left(-\frac{3}{2}\right)^{2}=5+\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}-3y+\frac{9}{4}=5+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
y^{2}-3y+\frac{9}{4}=\frac{29}{4}
Add 5 to \frac{9}{4}.
\left(y-\frac{3}{2}\right)^{2}=\frac{29}{4}
Factor y^{2}-3y+\frac{9}{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(y-\frac{3}{2}\right)^{2}}=\sqrt{\frac{29}{4}}
Take the square root of both sides of the equation.
y-\frac{3}{2}=\frac{\sqrt{29}}{2} y-\frac{3}{2}=-\frac{\sqrt{29}}{2}
Simplify.
y=\frac{\sqrt{29}+3}{2} y=\frac{3-\sqrt{29}}{2}
Add \frac{3}{2} to both sides of the equation.