Solve -27/4y+1+5=y

$$\frac { - 27 } { 4 y + 1 } + 5 = y$$

Solve for y

$$y=2$$

$$y = \frac{11}{4} = 2\frac{3}{4} = 2.75$$

Steps Using Factoring By Grouping

Variable y cannot be equal to -\frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by 4y+1.

$$-27+\left(4y+1\right)\times 5=y\left(4y+1\right)$$

Use the distributive property to multiply 4y+1 by 5.

$$-27+20y+5=y\left(4y+1\right)$$

Add -27 and 5 to get -22.

$$-22+20y=y\left(4y+1\right)$$

Use the distributive property to multiply y by 4y+1.

$$-22+20y=4y^{2}+y$$

Subtract 4y^{2} from both sides.

$$-22+20y-4y^{2}=y$$

Subtract y from both sides.

$$-22+20y-4y^{2}-y=0$$

Combine 20y and -y to get 19y.

$$-22+19y-4y^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-4y^{2}+19y-22=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4y^{2}+ay+by-22. To find a and b, set up a system to be solved.

$$a+b=19 ab=-4\left(-22\right)=88$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 88.

$$1,88 2,44 4,22 8,11$$

Calculate the sum for each pair.

$$1+88=89 2+44=46 4+22=26 8+11=19$$

The solution is the pair that gives sum 19.

$$a=11 b=8$$

Rewrite -4y^{2}+19y-22 as \left(-4y^{2}+11y\right)+\left(8y-22\right).

$$\left(-4y^{2}+11y\right)+\left(8y-22\right)$$

Factor out -y in the first and 2 in the second group.

$$-y\left(4y-11\right)+2\left(4y-11\right)$$

Factor out common term 4y-11 by using distributive property.

$$\left(4y-11\right)\left(-y+2\right)$$

To find equation solutions, solve 4y-11=0 and -y+2=0.

$$y=\frac{11}{4} y=2$$

Steps Using the Quadratic Formula

Variable y cannot be equal to -\frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by 4y+1.

$$-27+\left(4y+1\right)\times 5=y\left(4y+1\right)$$

Use the distributive property to multiply 4y+1 by 5.

$$-27+20y+5=y\left(4y+1\right)$$

Add -27 and 5 to get -22.

$$-22+20y=y\left(4y+1\right)$$

Use the distributive property to multiply y by 4y+1.

$$-22+20y=4y^{2}+y$$

Subtract 4y^{2} from both sides.

$$-22+20y-4y^{2}=y$$

Subtract y from both sides.

$$-22+20y-4y^{2}-y=0$$

Combine 20y and -y to get 19y.

$$-22+19y-4y^{2}=0$$

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.

$$-4y^{2}+19y-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 19 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-19±\sqrt{19^{2}-4\left(-4\right)\left(-22\right)}}{2\left(-4\right)}$$

Square 19.

$$y=\frac{-19±\sqrt{361-4\left(-4\right)\left(-22\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-19±\sqrt{361+16\left(-22\right)}}{2\left(-4\right)}$$

Multiply 16 times -22.

$$y=\frac{-19±\sqrt{361-352}}{2\left(-4\right)}$$

Add 361 to -352.

$$y=\frac{-19±\sqrt{9}}{2\left(-4\right)}$$

Take the square root of 9.

$$y=\frac{-19±3}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-19±3}{-8}$$

Now solve the equation y=\frac{-19±3}{-8} when ± is plus. Add -19 to 3.

$$y=-\frac{16}{-8}$$

Divide -16 by -8.

$$y=2$$

Now solve the equation y=\frac{-19±3}{-8} when ± is minus. Subtract 3 from -19.

$$y=-\frac{22}{-8}$$

Reduce the fraction \frac{-22}{-8} to lowest terms by extracting and canceling out 2.

$$y=\frac{11}{4}$$

The equation is now solved.

$$y=2 y=\frac{11}{4}$$

Steps for Completing the Square

Variable y cannot be equal to -\frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by 4y+1.

$$-27+\left(4y+1\right)\times 5=y\left(4y+1\right)$$

Use the distributive property to multiply 4y+1 by 5.

$$-27+20y+5=y\left(4y+1\right)$$

Add -27 and 5 to get -22.

$$-22+20y=y\left(4y+1\right)$$

Use the distributive property to multiply y by 4y+1.

$$-22+20y=4y^{2}+y$$

Subtract 4y^{2} from both sides.

$$-22+20y-4y^{2}=y$$

Subtract y from both sides.

$$-22+20y-4y^{2}-y=0$$

Combine 20y and -y to get 19y.

$$-22+19y-4y^{2}=0$$

Add 22 to both sides. Anything plus zero gives itself.

$$19y-4y^{2}=22$$

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.

$$-4y^{2}+19y=22$$

Divide both sides by -4.

$$\frac{-4y^{2}+19y}{-4}=\frac{22}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\frac{19}{-4}y=\frac{22}{-4}$$

Divide 19 by -4.

$$y^{2}-\frac{19}{4}y=\frac{22}{-4}$$

Reduce the fraction \frac{22}{-4} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{19}{4}y=-\frac{11}{2}$$

Divide -\frac{19}{4}, the coefficient of the x term, by 2 to get -\frac{19}{8}. Then add the square of -\frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{19}{4}y+\left(-\frac{19}{8}\right)^{2}=-\frac{11}{2}+\left(-\frac{19}{8}\right)^{2}$$

Square -\frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{19}{4}y+\frac{361}{64}=-\frac{11}{2}+\frac{361}{64}$$

Add -\frac{11}{2} to \frac{361}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{19}{4}y+\frac{361}{64}=\frac{9}{64}$$

Factor y^{2}-\frac{19}{4}y+\frac{361}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{19}{8}\right)^{2}=\frac{9}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{19}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Simplify.

$$y-\frac{19}{8}=\frac{3}{8} y-\frac{19}{8}=-\frac{3}{8}$$

Add \frac{19}{8} to both sides of the equation.

$$y=\frac{11}{4} y=2$$