Solve 11--4y-4/36y=47

$$\frac{ 11--4y-4 }{ 3 } 6y = 47$$

Solve for y

$$y = \frac{5 \sqrt{17} - 7}{8} \approx 1.701941016$$

$$y=\frac{-5\sqrt{17}-7}{8}\approx -3.451941016$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$\left(11-\left(-4y\right)-4\right)\times 6y=141$$

The opposite of -4y is 4y.

$$\left(11+4y-4\right)\times 6y=141$$

Subtract 4 from 11 to get 7.

$$\left(7+4y\right)\times 6y=141$$

Use the distributive property to multiply 7+4y by 6.

$$\left(42+24y\right)y=141$$

Use the distributive property to multiply 42+24y by y.

$$42y+24y^{2}=141$$

Subtract 141 from both sides.

$$42y+24y^{2}-141=0$$

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.

$$24y^{2}+42y-141=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 42 for b, and -141 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-42±\sqrt{42^{2}-4\times 24\left(-141\right)}}{2\times 24}$$

Square 42.

$$y=\frac{-42±\sqrt{1764-4\times 24\left(-141\right)}}{2\times 24}$$

Multiply -4 times 24.

$$y=\frac{-42±\sqrt{1764-96\left(-141\right)}}{2\times 24}$$

Multiply -96 times -141.

$$y=\frac{-42±\sqrt{1764+13536}}{2\times 24}$$

Add 1764 to 13536.

$$y=\frac{-42±\sqrt{15300}}{2\times 24}$$

Take the square root of 15300.

$$y=\frac{-42±30\sqrt{17}}{2\times 24}$$

Multiply 2 times 24.

$$y=\frac{-42±30\sqrt{17}}{48}$$

Now solve the equation y=\frac{-42±30\sqrt{17}}{48} when ± is plus. Add -42 to 30\sqrt{17}.

$$y=\frac{30\sqrt{17}-42}{48}$$

Divide -42+30\sqrt{17} by 48.

$$y=\frac{5\sqrt{17}-7}{8}$$

Now solve the equation y=\frac{-42±30\sqrt{17}}{48} when ± is minus. Subtract 30\sqrt{17} from -42.

$$y=\frac{-30\sqrt{17}-42}{48}$$

Divide -42-30\sqrt{17} by 48.

$$y=\frac{-5\sqrt{17}-7}{8}$$

The equation is now solved.

$$y=\frac{5\sqrt{17}-7}{8} y=\frac{-5\sqrt{17}-7}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$\left(11-\left(-4y\right)-4\right)\times 6y=141$$

The opposite of -4y is 4y.

$$\left(11+4y-4\right)\times 6y=141$$

Subtract 4 from 11 to get 7.

$$\left(7+4y\right)\times 6y=141$$

Use the distributive property to multiply 7+4y by 6.

$$\left(42+24y\right)y=141$$

Use the distributive property to multiply 42+24y by y.

$$42y+24y^{2}=141$$

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.

$$24y^{2}+42y=141$$

Divide both sides by 24.

$$\frac{24y^{2}+42y}{24}=\frac{141}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$y^{2}+\frac{42}{24}y=\frac{141}{24}$$

Reduce the fraction \frac{42}{24} to lowest terms by extracting and canceling out 6.

$$y^{2}+\frac{7}{4}y=\frac{141}{24}$$

Reduce the fraction \frac{141}{24} to lowest terms by extracting and canceling out 3.

$$y^{2}+\frac{7}{4}y=\frac{47}{8}$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{7}{4}y+\left(\frac{7}{8}\right)^{2}=\frac{47}{8}+\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{7}{4}y+\frac{49}{64}=\frac{47}{8}+\frac{49}{64}$$

Add \frac{47}{8} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{7}{4}y+\frac{49}{64}=\frac{425}{64}$$

Factor y^{2}+\frac{7}{4}y+\frac{49}{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}.

$$\left(y+\frac{7}{8}\right)^{2}=\frac{425}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{7}{8}\right)^{2}}=\sqrt{\frac{425}{64}}$$

Simplify.

$$y+\frac{7}{8}=\frac{5\sqrt{17}}{8} y+\frac{7}{8}=-\frac{5\sqrt{17}}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$y=\frac{5\sqrt{17}-7}{8} y=\frac{-5\sqrt{17}-7}{8}$$