Solve 101y^2+50y-52y=0

$$101 y ^ { 2 } + 50 y - 52 y = 0$$

Solve for y

$$y=\frac{2}{101}\approx 0.01980198$$

$$y=0$$

Steps Using Factoring

Factor out y.

$$y\left(101y+50-52\right)=0$$

To find equation solutions, solve y=0 and 101y-2=0.

$$y=0 y=\frac{2}{101}$$

Steps Using the Quadratic Formula

Combine 50y and -52y to get -2y.

$$101y^{2}-2y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 101 for a, -2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}}}{2\times 101}$$

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

$$y=\frac{-\left(-2\right)±2}{2\times 101}$$

The opposite of -2 is 2.

$$y=\frac{2±2}{2\times 101}$$

Multiply 2 times 101.

$$y=\frac{2±2}{202}$$

Now solve the equation y=\frac{2±2}{202} when ± is plus. Add 2 to 2.

$$y=\frac{4}{202}$$

Reduce the fraction \frac{4}{202} to lowest terms by extracting and canceling out 2.

$$y=\frac{2}{101}$$

Now solve the equation y=\frac{2±2}{202} when ± is minus. Subtract 2 from 2.

$$y=\frac{0}{202}$$

Divide 0 by 202.

$$y=0$$

The equation is now solved.

$$y=\frac{2}{101} y=0$$

Steps for Completing the Square

Combine 50y and -52y to get -2y.

$$101y^{2}-2y=0$$

Divide both sides by 101.

$$\frac{101y^{2}-2y}{101}=\frac{0}{101}$$

Dividing by 101 undoes the multiplication by 101.

$$y^{2}-\frac{2}{101}y=\frac{0}{101}$$

Divide 0 by 101.

$$y^{2}-\frac{2}{101}y=0$$

Divide -\frac{2}{101}, the coefficient of the x term, by 2 to get -\frac{1}{101}. Then add the square of -\frac{1}{101} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{2}{101}y+\left(-\frac{1}{101}\right)^{2}=\left(-\frac{1}{101}\right)^{2}$$

Square -\frac{1}{101} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{2}{101}y+\frac{1}{10201}=\frac{1}{10201}$$

Factor y^{2}-\frac{2}{101}y+\frac{1}{10201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{1}{101}\right)^{2}=\frac{1}{10201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{101}\right)^{2}}=\sqrt{\frac{1}{10201}}$$

Simplify.

$$y-\frac{1}{101}=\frac{1}{101} y-\frac{1}{101}=-\frac{1}{101}$$

Add \frac{1}{101} to both sides of the equation.

$$y=\frac{2}{101} y=0$$