Solve 5=-y^2+4y

$$5 = - y ^ { 2 } + 4 y$$

Solve for y

$$y=2+i$$

$$y=2-i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-y^{2}+4y=5$$

Subtract 5 from both sides.

$$-y^{2}+4y-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Square 4.

$$y=\frac{-4±\sqrt{16-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-4±\sqrt{16+4\left(-5\right)}}{2\left(-1\right)}$$

Multiply 4 times -5.

$$y=\frac{-4±\sqrt{16-20}}{2\left(-1\right)}$$

Add 16 to -20.

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

Take the square root of -4.

$$y=\frac{-4±2i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-4±2i}{-2}$$

Now solve the equation y=\frac{-4±2i}{-2} when ± is plus. Add -4 to 2i.

$$y=\frac{-4+2i}{-2}$$

Divide -4+2i by -2.

$$y=2-i$$

Now solve the equation y=\frac{-4±2i}{-2} when ± is minus. Subtract 2i from -4.

$$y=\frac{-4-2i}{-2}$$

Divide -4-2i by -2.

$$y=2+i$$

The equation is now solved.

$$y=2-i y=2+i$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-y^{2}+4y=5$$

Divide both sides by -1.

$$\frac{-y^{2}+4y}{-1}=\frac{5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{4}{-1}y=\frac{5}{-1}$$

Divide 4 by -1.

$$y^{2}-4y=\frac{5}{-1}$$

Divide 5 by -1.

$$y^{2}-4y=-5$$

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.

$$y^{2}-4y+\left(-2\right)^{2}=-5+\left(-2\right)^{2}$$

Square -2.

$$y^{2}-4y+4=-5+4$$

Add -5 to 4.

$$y^{2}-4y+4=-1$$

Factor y^{2}-4y+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}.

$$\left(y-2\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-2\right)^{2}}=\sqrt{-1}$$

Simplify.

$$y-2=i y-2=-i$$

Add 2 to both sides of the equation.

$$y=2+i y=2-i$$