Solve -18/4y+1+4=y

$$\frac { - 18 } { 4 y + 1 } + 4 = y$$

Solve for y

$$y=2$$

$$y = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

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.

$$-18+\left(4y+1\right)\times 4=y\left(4y+1\right)$$

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

$$-18+16y+4=y\left(4y+1\right)$$

Add -18 and 4 to get -14.

$$-14+16y=y\left(4y+1\right)$$

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

$$-14+16y=4y^{2}+y$$

Subtract 4y^{2} from both sides.

$$-14+16y-4y^{2}=y$$

Subtract y from both sides.

$$-14+16y-4y^{2}-y=0$$

Combine 16y and -y to get 15y.

$$-14+15y-4y^{2}=0$$

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

$$-4y^{2}+15y-14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4y^{2}+ay+by-14. To find a and b, set up a system to be solved.

$$a+b=15 ab=-4\left(-14\right)=56$$

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 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 15.

$$a=8 b=7$$

Rewrite -4y^{2}+15y-14 as \left(-4y^{2}+8y\right)+\left(7y-14\right).

$$\left(-4y^{2}+8y\right)+\left(7y-14\right)$$

Factor out 4y in the first and -7 in the second group.

$$4y\left(-y+2\right)-7\left(-y+2\right)$$

Factor out common term -y+2 by using distributive property.

$$\left(-y+2\right)\left(4y-7\right)$$

To find equation solutions, solve -y+2=0 and 4y-7=0.

$$y=2 y=\frac{7}{4}$$

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.

$$-18+\left(4y+1\right)\times 4=y\left(4y+1\right)$$

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

$$-18+16y+4=y\left(4y+1\right)$$

Add -18 and 4 to get -14.

$$-14+16y=y\left(4y+1\right)$$

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

$$-14+16y=4y^{2}+y$$

Subtract 4y^{2} from both sides.

$$-14+16y-4y^{2}=y$$

Subtract y from both sides.

$$-14+16y-4y^{2}-y=0$$

Combine 16y and -y to get 15y.

$$-14+15y-4y^{2}=0$$

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.

$$-4y^{2}+15y-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 15 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-15±\sqrt{15^{2}-4\left(-4\right)\left(-14\right)}}{2\left(-4\right)}$$

Square 15.

$$y=\frac{-15±\sqrt{225-4\left(-4\right)\left(-14\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-15±\sqrt{225+16\left(-14\right)}}{2\left(-4\right)}$$

Multiply 16 times -14.

$$y=\frac{-15±\sqrt{225-224}}{2\left(-4\right)}$$

Add 225 to -224.

$$y=\frac{-15±\sqrt{1}}{2\left(-4\right)}$$

Take the square root of 1.

$$y=\frac{-15±1}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-15±1}{-8}$$

Now solve the equation y=\frac{-15±1}{-8} when ± is plus. Add -15 to 1.

$$y=-\frac{14}{-8}$$

Reduce the fraction \frac{-14}{-8} to lowest terms by extracting and canceling out 2.

$$y=\frac{7}{4}$$

Now solve the equation y=\frac{-15±1}{-8} when ± is minus. Subtract 1 from -15.

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

Divide -16 by -8.

$$y=2$$

The equation is now solved.

$$y=\frac{7}{4} y=2$$

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.

$$-18+\left(4y+1\right)\times 4=y\left(4y+1\right)$$

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

$$-18+16y+4=y\left(4y+1\right)$$

Add -18 and 4 to get -14.

$$-14+16y=y\left(4y+1\right)$$

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

$$-14+16y=4y^{2}+y$$

Subtract 4y^{2} from both sides.

$$-14+16y-4y^{2}=y$$

Subtract y from both sides.

$$-14+16y-4y^{2}-y=0$$

Combine 16y and -y to get 15y.

$$-14+15y-4y^{2}=0$$

Add 14 to both sides. Anything plus zero gives itself.

$$15y-4y^{2}=14$$

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.

$$-4y^{2}+15y=14$$

Divide both sides by -4.

$$\frac{-4y^{2}+15y}{-4}=\frac{14}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\frac{15}{-4}y=\frac{14}{-4}$$

Divide 15 by -4.

$$y^{2}-\frac{15}{4}y=\frac{14}{-4}$$

Reduce the fraction \frac{14}{-4} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{15}{4}y=-\frac{7}{2}$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{15}{4}y+\left(-\frac{15}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{15}{4}y+\frac{225}{64}=-\frac{7}{2}+\frac{225}{64}$$

Add -\frac{7}{2} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{15}{4}y+\frac{225}{64}=\frac{1}{64}$$

Factor y^{2}-\frac{15}{4}y+\frac{225}{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}.

$$\left(y-\frac{15}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{15}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$y-\frac{15}{8}=\frac{1}{8} y-\frac{15}{8}=-\frac{1}{8}$$

Add \frac{15}{8} to both sides of the equation.

$$y=2 y=\frac{7}{4}$$