Solve (55-y)*y=2y

$$( 55 - y ) \cdot y = 2 y$$

Solve for y

$$y=53$$

$$y=0$$

Steps Using Factoring

Use the distributive property to multiply 55-y by y.

$$55y-y^{2}=2y$$

Subtract 2y from both sides.

$$55y-y^{2}-2y=0$$

Combine 55y and -2y to get 53y.

$$53y-y^{2}=0$$

Factor out y.

$$y\left(53-y\right)=0$$

To find equation solutions, solve y=0 and 53-y=0.

$$y=0 y=53$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 55-y by y.

$$55y-y^{2}=2y$$

Subtract 2y from both sides.

$$55y-y^{2}-2y=0$$

Combine 55y and -2y to get 53y.

$$53y-y^{2}=0$$

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.

$$-y^{2}+53y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 53 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-53±\sqrt{53^{2}}}{2\left(-1\right)}$$

Take the square root of 53^{2}.

$$y=\frac{-53±53}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-53±53}{-2}$$

Now solve the equation y=\frac{-53±53}{-2} when ± is plus. Add -53 to 53.

$$y=\frac{0}{-2}$$

Divide 0 by -2.

$$y=0$$

Now solve the equation y=\frac{-53±53}{-2} when ± is minus. Subtract 53 from -53.

$$y=-\frac{106}{-2}$$

Divide -106 by -2.

$$y=53$$

The equation is now solved.

$$y=0 y=53$$

Steps for Completing the Square

Use the distributive property to multiply 55-y by y.

$$55y-y^{2}=2y$$

Subtract 2y from both sides.

$$55y-y^{2}-2y=0$$

Combine 55y and -2y to get 53y.

$$53y-y^{2}=0$$

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.

$$-y^{2}+53y=0$$

Divide both sides by -1.

$$\frac{-y^{2}+53y}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{53}{-1}y=\frac{0}{-1}$$

Divide 53 by -1.

$$y^{2}-53y=\frac{0}{-1}$$

Divide 0 by -1.

$$y^{2}-53y=0$$

Divide -53, the coefficient of the x term, by 2 to get -\frac{53}{2}. Then add the square of -\frac{53}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-53y+\left(-\frac{53}{2}\right)^{2}=\left(-\frac{53}{2}\right)^{2}$$

Square -\frac{53}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-53y+\frac{2809}{4}=\frac{2809}{4}$$

Factor y^{2}-53y+\frac{2809}{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}.

$$\left(y-\frac{53}{2}\right)^{2}=\frac{2809}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{53}{2}\right)^{2}}=\sqrt{\frac{2809}{4}}$$

Simplify.

$$y-\frac{53}{2}=\frac{53}{2} y-\frac{53}{2}=-\frac{53}{2}$$

Add \frac{53}{2} to both sides of the equation.

$$y=53 y=0$$