Solve y+10y-13/16y^2-9=3y-7/3-4y+6y+5/3+4y

$$\frac { y + 10 y - 13 } { 16 y ^ { 2 } - 9 } = \frac { 3 y - 7 } { 3 - 4 y } + \frac { 6 y + 5 } { 3 + 4 y }$$

Solve for y

$$y=\frac{-5+\sqrt{203}i}{12}\approx -0.416666667+1.187317237i$$

$$y=\frac{-\sqrt{203}i-5}{12}\approx -0.416666667-1.187317237i$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -\frac{3}{4},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4y-3\right)\left(4y+3\right), the least common multiple of 16y^{2}-9,3-4y,3+4y.

$$y+10y-13=\left(-3-4y\right)\left(3y-7\right)+\left(4y-3\right)\left(6y+5\right)$$

Combine y and 10y to get 11y.

$$11y-13=\left(-3-4y\right)\left(3y-7\right)+\left(4y-3\right)\left(6y+5\right)$$

Use the distributive property to multiply -3-4y by 3y-7 and combine like terms.

$$11y-13=19y+21-12y^{2}+\left(4y-3\right)\left(6y+5\right)$$

Use the distributive property to multiply 4y-3 by 6y+5 and combine like terms.

$$11y-13=19y+21-12y^{2}+24y^{2}+2y-15$$

Combine -12y^{2} and 24y^{2} to get 12y^{2}.

$$11y-13=19y+21+12y^{2}+2y-15$$

Combine 19y and 2y to get 21y.

$$11y-13=21y+21+12y^{2}-15$$

Subtract 15 from 21 to get 6.

$$11y-13=21y+6+12y^{2}$$

Subtract 21y from both sides.

$$11y-13-21y=6+12y^{2}$$

Combine 11y and -21y to get -10y.

$$-10y-13=6+12y^{2}$$

Subtract 6 from both sides.

$$-10y-13-6=12y^{2}$$

Subtract 6 from -13 to get -19.

$$-10y-19=12y^{2}$$

Subtract 12y^{2} from both sides.

$$-10y-19-12y^{2}=0$$

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.

$$-12y^{2}-10y-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -10 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-12\right)\left(-19\right)}}{2\left(-12\right)}$$

Square -10.

$$y=\frac{-\left(-10\right)±\sqrt{100-4\left(-12\right)\left(-19\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$y=\frac{-\left(-10\right)±\sqrt{100+48\left(-19\right)}}{2\left(-12\right)}$$

Multiply 48 times -19.

$$y=\frac{-\left(-10\right)±\sqrt{100-912}}{2\left(-12\right)}$$

Add 100 to -912.

$$y=\frac{-\left(-10\right)±\sqrt{-812}}{2\left(-12\right)}$$

Take the square root of -812.

$$y=\frac{-\left(-10\right)±2\sqrt{203}i}{2\left(-12\right)}$$

The opposite of -10 is 10.

$$y=\frac{10±2\sqrt{203}i}{2\left(-12\right)}$$

Multiply 2 times -12.

$$y=\frac{10±2\sqrt{203}i}{-24}$$

Now solve the equation y=\frac{10±2\sqrt{203}i}{-24} when ± is plus. Add 10 to 2i\sqrt{203}.

$$y=\frac{10+2\sqrt{203}i}{-24}$$

Divide 10+2i\sqrt{203} by -24.

$$y=\frac{-\sqrt{203}i-5}{12}$$

Now solve the equation y=\frac{10±2\sqrt{203}i}{-24} when ± is minus. Subtract 2i\sqrt{203} from 10.

$$y=\frac{-2\sqrt{203}i+10}{-24}$$

Divide 10-2i\sqrt{203} by -24.

$$y=\frac{-5+\sqrt{203}i}{12}$$

The equation is now solved.

$$y=\frac{-\sqrt{203}i-5}{12} y=\frac{-5+\sqrt{203}i}{12}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -\frac{3}{4},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4y-3\right)\left(4y+3\right), the least common multiple of 16y^{2}-9,3-4y,3+4y.

$$y+10y-13=\left(-3-4y\right)\left(3y-7\right)+\left(4y-3\right)\left(6y+5\right)$$

Combine y and 10y to get 11y.

$$11y-13=\left(-3-4y\right)\left(3y-7\right)+\left(4y-3\right)\left(6y+5\right)$$

Use the distributive property to multiply -3-4y by 3y-7 and combine like terms.

$$11y-13=19y+21-12y^{2}+\left(4y-3\right)\left(6y+5\right)$$

Use the distributive property to multiply 4y-3 by 6y+5 and combine like terms.

$$11y-13=19y+21-12y^{2}+24y^{2}+2y-15$$

Combine -12y^{2} and 24y^{2} to get 12y^{2}.

$$11y-13=19y+21+12y^{2}+2y-15$$

Combine 19y and 2y to get 21y.

$$11y-13=21y+21+12y^{2}-15$$

Subtract 15 from 21 to get 6.

$$11y-13=21y+6+12y^{2}$$

Subtract 21y from both sides.

$$11y-13-21y=6+12y^{2}$$

Combine 11y and -21y to get -10y.

$$-10y-13=6+12y^{2}$$

Subtract 12y^{2} from both sides.

$$-10y-13-12y^{2}=6$$

Add 13 to both sides.

$$-10y-12y^{2}=6+13$$

Add 6 and 13 to get 19.

$$-10y-12y^{2}=19$$

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.

$$-12y^{2}-10y=19$$

Divide both sides by -12.

$$\frac{-12y^{2}-10y}{-12}=\frac{19}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$y^{2}+\left(-\frac{10}{-12}\right)y=\frac{19}{-12}$$

Reduce the fraction \frac{-10}{-12} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{5}{6}y=\frac{19}{-12}$$

Divide 19 by -12.

$$y^{2}+\frac{5}{6}y=-\frac{19}{12}$$

Divide \frac{5}{6}, the coefficient of the x term, by 2 to get \frac{5}{12}. Then add the square of \frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{5}{6}y+\left(\frac{5}{12}\right)^{2}=-\frac{19}{12}+\left(\frac{5}{12}\right)^{2}$$

Square \frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{5}{6}y+\frac{25}{144}=-\frac{19}{12}+\frac{25}{144}$$

Add -\frac{19}{12} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{5}{6}y+\frac{25}{144}=-\frac{203}{144}$$

Factor y^{2}+\frac{5}{6}y+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{5}{12}\right)^{2}=-\frac{203}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{5}{12}\right)^{2}}=\sqrt{-\frac{203}{144}}$$

Simplify.

$$y+\frac{5}{12}=\frac{\sqrt{203}i}{12} y+\frac{5}{12}=-\frac{\sqrt{203}i}{12}$$

Subtract \frac{5}{12} from both sides of the equation.

$$y=\frac{-5+\sqrt{203}i}{12} y=\frac{-\sqrt{203}i-5}{12}$$