Solve 2y^2=5y

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

Solve for y

$$y = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$y=0$$

Steps Using Factoring

Subtract 5y from both sides.

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

Factor out y.

$$y\left(2y-5\right)=0$$

To find equation solutions, solve y=0 and 2y-5=0.

$$y=0 y=\frac{5}{2}$$

Steps Using the Quadratic Formula

Subtract 5y from both sides.

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

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

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

Take the square root of \left(-5\right)^{2}.

$$y=\frac{-\left(-5\right)±5}{2\times 2}$$

The opposite of -5 is 5.

$$y=\frac{5±5}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{5±5}{4}$$

Now solve the equation y=\frac{5±5}{4} when ± is plus. Add 5 to 5.

$$y=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$y=\frac{5}{2}$$

Now solve the equation y=\frac{5±5}{4} when ± is minus. Subtract 5 from 5.

$$y=\frac{0}{4}$$

Divide 0 by 4.

$$y=0$$

The equation is now solved.

$$y=\frac{5}{2} y=0$$

Steps for Completing the Square

Subtract 5y from both sides.

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

Divide both sides by 2.

$$\frac{2y^{2}-5y}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$y^{2}-\frac{5}{2}y=\frac{0}{2}$$

Divide 0 by 2.

$$y^{2}-\frac{5}{2}y=0$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{5}{2}y+\frac{25}{16}=\frac{25}{16}$$

Factor y^{2}-\frac{5}{2}y+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{5}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{5}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$y-\frac{5}{4}=\frac{5}{4} y-\frac{5}{4}=-\frac{5}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$y=\frac{5}{2} y=0$$