Solve 77+1/3-5-2y/2y-4y-3/4=1+4y^2/3y

$$\frac { 77 + 1 } { 3 } - \frac { 5 - 2 y } { 2 y } - \frac { 4 y - 3 } { 4 } = \frac { 1 + 4 y ^ { 2 } } { 3 y }$$

Solve for y

$$y = \frac{\sqrt{107081} + 333}{56} \approx 11.789863096$$

$$y=\frac{333-\sqrt{107081}}{56}\approx 0.102994047$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12y, the least common multiple of 3,2y,4,3y.

$$4y\left(77+1\right)-6\left(5-2y\right)-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Add 77 and 1 to get 78.

$$4y\times 78-6\left(5-2y\right)-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Multiply 4 and 78 to get 312.

$$312y-6\left(5-2y\right)-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Use the distributive property to multiply -6 by 5-2y.

$$312y-30+12y-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Combine 312y and 12y to get 324y.

$$324y-30-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Use the distributive property to multiply 3y by 4y-3.

$$324y-30-\left(12y^{2}-9y\right)=4\left(1+4y^{2}\right)$$

To find the opposite of 12y^{2}-9y, find the opposite of each term.

$$324y-30-12y^{2}+9y=4\left(1+4y^{2}\right)$$

Combine 324y and 9y to get 333y.

$$333y-30-12y^{2}=4\left(1+4y^{2}\right)$$

Use the distributive property to multiply 4 by 1+4y^{2}.

$$333y-30-12y^{2}=4+16y^{2}$$

Subtract 4 from both sides.

$$333y-30-12y^{2}-4=16y^{2}$$

Subtract 4 from -30 to get -34.

$$333y-34-12y^{2}=16y^{2}$$

Subtract 16y^{2} from both sides.

$$333y-34-12y^{2}-16y^{2}=0$$

Combine -12y^{2} and -16y^{2} to get -28y^{2}.

$$333y-34-28y^{2}=0$$

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.

$$-28y^{2}+333y-34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -28 for a, 333 for b, and -34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-333±\sqrt{333^{2}-4\left(-28\right)\left(-34\right)}}{2\left(-28\right)}$$

Square 333.

$$y=\frac{-333±\sqrt{110889-4\left(-28\right)\left(-34\right)}}{2\left(-28\right)}$$

Multiply -4 times -28.

$$y=\frac{-333±\sqrt{110889+112\left(-34\right)}}{2\left(-28\right)}$$

Multiply 112 times -34.

$$y=\frac{-333±\sqrt{110889-3808}}{2\left(-28\right)}$$

Add 110889 to -3808.

$$y=\frac{-333±\sqrt{107081}}{2\left(-28\right)}$$

Multiply 2 times -28.

$$y=\frac{-333±\sqrt{107081}}{-56}$$

Now solve the equation y=\frac{-333±\sqrt{107081}}{-56} when ± is plus. Add -333 to \sqrt{107081}.

$$y=\frac{\sqrt{107081}-333}{-56}$$

Divide -333+\sqrt{107081} by -56.

$$y=\frac{333-\sqrt{107081}}{56}$$

Now solve the equation y=\frac{-333±\sqrt{107081}}{-56} when ± is minus. Subtract \sqrt{107081} from -333.

$$y=\frac{-\sqrt{107081}-333}{-56}$$

Divide -333-\sqrt{107081} by -56.

$$y=\frac{\sqrt{107081}+333}{56}$$

The equation is now solved.

$$y=\frac{333-\sqrt{107081}}{56} y=\frac{\sqrt{107081}+333}{56}$$

Steps for Completing the Square

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12y, the least common multiple of 3,2y,4,3y.

$$4y\left(77+1\right)-6\left(5-2y\right)-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Add 77 and 1 to get 78.

$$4y\times 78-6\left(5-2y\right)-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Multiply 4 and 78 to get 312.

$$312y-6\left(5-2y\right)-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Use the distributive property to multiply -6 by 5-2y.

$$312y-30+12y-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Combine 312y and 12y to get 324y.

$$324y-30-3y\left(4y-3\right)=4\left(1+4y^{2}\right)$$

Use the distributive property to multiply 3y by 4y-3.

$$324y-30-\left(12y^{2}-9y\right)=4\left(1+4y^{2}\right)$$

To find the opposite of 12y^{2}-9y, find the opposite of each term.

$$324y-30-12y^{2}+9y=4\left(1+4y^{2}\right)$$

Combine 324y and 9y to get 333y.

$$333y-30-12y^{2}=4\left(1+4y^{2}\right)$$

Use the distributive property to multiply 4 by 1+4y^{2}.

$$333y-30-12y^{2}=4+16y^{2}$$

Subtract 16y^{2} from both sides.

$$333y-30-12y^{2}-16y^{2}=4$$

Combine -12y^{2} and -16y^{2} to get -28y^{2}.

$$333y-30-28y^{2}=4$$

Add 30 to both sides.

$$333y-28y^{2}=4+30$$

Add 4 and 30 to get 34.

$$333y-28y^{2}=34$$

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.

$$-28y^{2}+333y=34$$

Divide both sides by -28.

$$\frac{-28y^{2}+333y}{-28}=\frac{34}{-28}$$

Dividing by -28 undoes the multiplication by -28.

$$y^{2}+\frac{333}{-28}y=\frac{34}{-28}$$

Divide 333 by -28.

$$y^{2}-\frac{333}{28}y=\frac{34}{-28}$$

Reduce the fraction \frac{34}{-28} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{333}{28}y=-\frac{17}{14}$$

Divide -\frac{333}{28}, the coefficient of the x term, by 2 to get -\frac{333}{56}. Then add the square of -\frac{333}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{333}{28}y+\left(-\frac{333}{56}\right)^{2}=-\frac{17}{14}+\left(-\frac{333}{56}\right)^{2}$$

Square -\frac{333}{56} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{333}{28}y+\frac{110889}{3136}=-\frac{17}{14}+\frac{110889}{3136}$$

Add -\frac{17}{14} to \frac{110889}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{333}{28}y+\frac{110889}{3136}=\frac{107081}{3136}$$

Factor y^{2}-\frac{333}{28}y+\frac{110889}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{333}{56}\right)^{2}=\frac{107081}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{333}{56}\right)^{2}}=\sqrt{\frac{107081}{3136}}$$

Simplify.

$$y-\frac{333}{56}=\frac{\sqrt{107081}}{56} y-\frac{333}{56}=-\frac{\sqrt{107081}}{56}$$

Add \frac{333}{56} to both sides of the equation.

$$y=\frac{\sqrt{107081}+333}{56} y=\frac{333-\sqrt{107081}}{56}$$