Solve -12/5y^2+y=0

$$- 1 \frac { 2 } { 5 } y ^ { 2 } + y = 0$$

Solve for y

$$y=\frac{5}{7}\approx 0.714285714$$

$$y=0$$

Steps Using Factoring

Factor out y.

$$y\left(\left(-\frac{5+2}{5}\right)y+1\right)=0$$

To find equation solutions, solve y=0 and -\frac{7}{5}y+1=0.

$$y=0 y=\frac{5}{7}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$5\left(-\frac{1\times 5+2}{5}\right)y^{2}+5y=0$$

Multiply 1 and 5 to get 5.

$$5\left(-\frac{5+2}{5}\right)y^{2}+5y=0$$

Add 5 and 2 to get 7.

$$5\left(-\frac{7}{5}\right)y^{2}+5y=0$$

Multiply 5 and -\frac{7}{5} to get -7.

$$-7y^{2}+5y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-5±\sqrt{5^{2}}}{2\left(-7\right)}$$

Take the square root of 5^{2}.

$$y=\frac{-5±5}{2\left(-7\right)}$$

Multiply 2 times -7.

$$y=\frac{-5±5}{-14}$$

Now solve the equation y=\frac{-5±5}{-14} when ± is plus. Add -5 to 5.

$$y=\frac{0}{-14}$$

Divide 0 by -14.

$$y=0$$

Now solve the equation y=\frac{-5±5}{-14} when ± is minus. Subtract 5 from -5.

$$y=-\frac{10}{-14}$$

Reduce the fraction \frac{-10}{-14} to lowest terms by extracting and canceling out 2.

$$y=\frac{5}{7}$$

The equation is now solved.

$$y=0 y=\frac{5}{7}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$5\left(-\frac{1\times 5+2}{5}\right)y^{2}+5y=0$$

Multiply 1 and 5 to get 5.

$$5\left(-\frac{5+2}{5}\right)y^{2}+5y=0$$

Add 5 and 2 to get 7.

$$5\left(-\frac{7}{5}\right)y^{2}+5y=0$$

Multiply 5 and -\frac{7}{5} to get -7.

$$-7y^{2}+5y=0$$

Divide both sides by -7.

$$\frac{-7y^{2}+5y}{-7}=\frac{0}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$y^{2}+\frac{5}{-7}y=\frac{0}{-7}$$

Divide 5 by -7.

$$y^{2}-\frac{5}{7}y=\frac{0}{-7}$$

Divide 0 by -7.

$$y^{2}-\frac{5}{7}y=0$$

Divide -\frac{5}{7}, the coefficient of the x term, by 2 to get -\frac{5}{14}. Then add the square of -\frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{5}{7}y+\left(-\frac{5}{14}\right)^{2}=\left(-\frac{5}{14}\right)^{2}$$

Square -\frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{5}{7}y+\frac{25}{196}=\frac{25}{196}$$

Factor y^{2}-\frac{5}{7}y+\frac{25}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{5}{14}\right)^{2}=\frac{25}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{5}{14}\right)^{2}}=\sqrt{\frac{25}{196}}$$

Simplify.

$$y-\frac{5}{14}=\frac{5}{14} y-\frac{5}{14}=-\frac{5}{14}$$

Add \frac{5}{14} to both sides of the equation.

$$y=\frac{5}{7} y=0$$