Solve 6y^2+3y=5y-3

$$6 y ^ { 2 } + 3 y = 5 y - 3$$

Solve for y

$$y=\frac{1+\sqrt{17}i}{6}\approx 0.166666667+0.687184271i$$

$$y=\frac{-\sqrt{17}i+1}{6}\approx 0.166666667-0.687184271i$$

Steps Using the Quadratic Formula

Subtract 5y from both sides.

$$6y^{2}+3y-5y=-3$$

Combine 3y and -5y to get -2y.

$$6y^{2}-2y=-3$$

Add 3 to both sides.

$$6y^{2}-2y+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -2 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 6\times 3}}{2\times 6}$$

Square -2.

$$y=\frac{-\left(-2\right)±\sqrt{4-4\times 6\times 3}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-\left(-2\right)±\sqrt{4-24\times 3}}{2\times 6}$$

Multiply -24 times 3.

$$y=\frac{-\left(-2\right)±\sqrt{4-72}}{2\times 6}$$

Add 4 to -72.

$$y=\frac{-\left(-2\right)±\sqrt{-68}}{2\times 6}$$

Take the square root of -68.

$$y=\frac{-\left(-2\right)±2\sqrt{17}i}{2\times 6}$$

The opposite of -2 is 2.

$$y=\frac{2±2\sqrt{17}i}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{2±2\sqrt{17}i}{12}$$

Now solve the equation y=\frac{2±2\sqrt{17}i}{12} when ± is plus. Add 2 to 2i\sqrt{17}.

$$y=\frac{2+2\sqrt{17}i}{12}$$

Divide 2+2i\sqrt{17} by 12.

$$y=\frac{1+\sqrt{17}i}{6}$$

Now solve the equation y=\frac{2±2\sqrt{17}i}{12} when ± is minus. Subtract 2i\sqrt{17} from 2.

$$y=\frac{-2\sqrt{17}i+2}{12}$$

Divide 2-2i\sqrt{17} by 12.

$$y=\frac{-\sqrt{17}i+1}{6}$$

The equation is now solved.

$$y=\frac{1+\sqrt{17}i}{6} y=\frac{-\sqrt{17}i+1}{6}$$

Steps for Completing the Square

Subtract 5y from both sides.

$$6y^{2}+3y-5y=-3$$

Combine 3y and -5y to get -2y.

$$6y^{2}-2y=-3$$

Divide both sides by 6.

$$\frac{6y^{2}-2y}{6}=-\frac{3}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$y^{2}+\left(-\frac{2}{6}\right)y=-\frac{3}{6}$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{1}{3}y=-\frac{3}{6}$$

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

$$y^{2}-\frac{1}{3}y=-\frac{1}{2}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{1}{3}y+\left(-\frac{1}{6}\right)^{2}=-\frac{1}{2}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{1}{3}y+\frac{1}{36}=-\frac{1}{2}+\frac{1}{36}$$

Add -\frac{1}{2} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{1}{3}y+\frac{1}{36}=-\frac{17}{36}$$

Factor y^{2}-\frac{1}{3}y+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{1}{6}\right)^{2}=-\frac{17}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{6}\right)^{2}}=\sqrt{-\frac{17}{36}}$$

Simplify.

$$y-\frac{1}{6}=\frac{\sqrt{17}i}{6} y-\frac{1}{6}=-\frac{\sqrt{17}i}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$y=\frac{1+\sqrt{17}i}{6} y=\frac{-\sqrt{17}i+1}{6}$$