Solve (31+24y/14)^2+y^2-8(31+24y/14)+14y=128

$$( \frac { 31 + 24 y } { 14 } ) ^ { 2 } + y ^ { 2 } - 8 ( \frac { 31 + 24 y } { 14 } ) + 14 y = 128$$

Solve for y

$$y=\frac{7\sqrt{3}}{2}-1\approx 5.062177826$$

$$y=-\frac{7\sqrt{3}}{2}-1\approx -7.062177826$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 14.

$$14\left(\left(\frac{31+24y}{14}\right)^{2}+y^{2}-8\times \frac{31+24y}{14}\right)+196y=1792$$

To raise \frac{31+24y}{14} to a power, raise both numerator and denominator to the power and then divide.

$$14\left(\frac{\left(31+24y\right)^{2}}{14^{2}}+y^{2}-8\times \frac{31+24y}{14}\right)+196y=1792$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{2} times \frac{14^{2}}{14^{2}}.

$$14\left(\frac{\left(31+24y\right)^{2}}{14^{2}}+\frac{y^{2}\times 14^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Since \frac{\left(31+24y\right)^{2}}{14^{2}} and \frac{y^{2}\times 14^{2}}{14^{2}} have the same denominator, add them by adding their numerators.

$$14\left(\frac{\left(31+24y\right)^{2}+y^{2}\times 14^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Do the multiplications in \left(31+24y\right)^{2}+y^{2}\times 14^{2}.

$$14\left(\frac{961+1488y+576y^{2}+196y^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Combine like terms in 961+1488y+576y^{2}+196y^{2}.

$$14\left(\frac{961+1488y+772y^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Express 8\times \frac{31+24y}{14} as a single fraction.

$$14\left(\frac{961+1488y+772y^{2}}{14^{2}}-\frac{8\left(31+24y\right)}{14}\right)+196y=1792$$

Use the distributive property to multiply 8 by 31+24y.

$$14\left(\frac{961+1488y+772y^{2}}{14^{2}}-\frac{248+192y}{14}\right)+196y=1792$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 14^{2} and 14 is 196. Multiply \frac{248+192y}{14} times \frac{14}{14}.

$$14\left(\frac{961+1488y+772y^{2}}{196}-\frac{14\left(248+192y\right)}{196}\right)+196y=1792$$

Since \frac{961+1488y+772y^{2}}{196} and \frac{14\left(248+192y\right)}{196} have the same denominator, subtract them by subtracting their numerators.

$$14\times \frac{961+1488y+772y^{2}-14\left(248+192y\right)}{196}+196y=1792$$

Do the multiplications in 961+1488y+772y^{2}-14\left(248+192y\right).

$$14\times \frac{961+1488y+772y^{2}-3472-2688y}{196}+196y=1792$$

Combine like terms in 961+1488y+772y^{2}-3472-2688y.

$$14\times \frac{-2511-1200y+772y^{2}}{196}+196y=1792$$

Cancel out 196, the greatest common factor in 14 and 196.

$$\frac{-2511-1200y+772y^{2}}{14}+196y=1792$$

Divide each term of -2511-1200y+772y^{2} by 14 to get -\frac{2511}{14}-\frac{600}{7}y+\frac{386}{7}y^{2}.

$$-\frac{2511}{14}-\frac{600}{7}y+\frac{386}{7}y^{2}+196y=1792$$

Combine -\frac{600}{7}y and 196y to get \frac{772}{7}y.

$$-\frac{2511}{14}+\frac{772}{7}y+\frac{386}{7}y^{2}=1792$$

Subtract 1792 from both sides.

$$-\frac{2511}{14}+\frac{772}{7}y+\frac{386}{7}y^{2}-1792=0$$

Subtract 1792 from -\frac{2511}{14} to get -\frac{27599}{14}.

$$-\frac{27599}{14}+\frac{772}{7}y+\frac{386}{7}y^{2}=0$$

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.

$$\frac{386}{7}y^{2}+\frac{772}{7}y-\frac{27599}{14}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{386}{7} for a, \frac{772}{7} for b, and -\frac{27599}{14} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{772}{7}±\sqrt{\left(\frac{772}{7}\right)^{2}-4\times \frac{386}{7}\left(-\frac{27599}{14}\right)}}{2\times \frac{386}{7}}$$

Square \frac{772}{7} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\frac{772}{7}±\sqrt{\frac{595984}{49}-4\times \frac{386}{7}\left(-\frac{27599}{14}\right)}}{2\times \frac{386}{7}}$$

Multiply -4 times \frac{386}{7}.

$$y=\frac{-\frac{772}{7}±\sqrt{\frac{595984}{49}-\frac{1544}{7}\left(-\frac{27599}{14}\right)}}{2\times \frac{386}{7}}$$

Multiply -\frac{1544}{7} times -\frac{27599}{14} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{772}{7}±\sqrt{\frac{595984+21306428}{49}}}{2\times \frac{386}{7}}$$

Add \frac{595984}{49} to \frac{21306428}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{772}{7}±\sqrt{446988}}{2\times \frac{386}{7}}$$

Take the square root of 446988.

$$y=\frac{-\frac{772}{7}±386\sqrt{3}}{2\times \frac{386}{7}}$$

Multiply 2 times \frac{386}{7}.

$$y=\frac{-\frac{772}{7}±386\sqrt{3}}{\frac{772}{7}}$$

Now solve the equation y=\frac{-\frac{772}{7}±386\sqrt{3}}{\frac{772}{7}} when ± is plus. Add -\frac{772}{7} to 386\sqrt{3}.

$$y=\frac{386\sqrt{3}-\frac{772}{7}}{\frac{772}{7}}$$

Divide -\frac{772}{7}+386\sqrt{3} by \frac{772}{7} by multiplying -\frac{772}{7}+386\sqrt{3} by the reciprocal of \frac{772}{7}.

$$y=\frac{7\sqrt{3}}{2}-1$$

Now solve the equation y=\frac{-\frac{772}{7}±386\sqrt{3}}{\frac{772}{7}} when ± is minus. Subtract 386\sqrt{3} from -\frac{772}{7}.

$$y=\frac{-386\sqrt{3}-\frac{772}{7}}{\frac{772}{7}}$$

Divide -\frac{772}{7}-386\sqrt{3} by \frac{772}{7} by multiplying -\frac{772}{7}-386\sqrt{3} by the reciprocal of \frac{772}{7}.

$$y=-\frac{7\sqrt{3}}{2}-1$$

The equation is now solved.

$$y=\frac{7\sqrt{3}}{2}-1 y=-\frac{7\sqrt{3}}{2}-1$$

Steps for Completing the Square

Multiply both sides of the equation by 14.

$$14\left(\left(\frac{31+24y}{14}\right)^{2}+y^{2}-8\times \frac{31+24y}{14}\right)+196y=1792$$

To raise \frac{31+24y}{14} to a power, raise both numerator and denominator to the power and then divide.

$$14\left(\frac{\left(31+24y\right)^{2}}{14^{2}}+y^{2}-8\times \frac{31+24y}{14}\right)+196y=1792$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{2} times \frac{14^{2}}{14^{2}}.

$$14\left(\frac{\left(31+24y\right)^{2}}{14^{2}}+\frac{y^{2}\times 14^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Since \frac{\left(31+24y\right)^{2}}{14^{2}} and \frac{y^{2}\times 14^{2}}{14^{2}} have the same denominator, add them by adding their numerators.

$$14\left(\frac{\left(31+24y\right)^{2}+y^{2}\times 14^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Do the multiplications in \left(31+24y\right)^{2}+y^{2}\times 14^{2}.

$$14\left(\frac{961+1488y+576y^{2}+196y^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Combine like terms in 961+1488y+576y^{2}+196y^{2}.

$$14\left(\frac{961+1488y+772y^{2}}{14^{2}}-8\times \frac{31+24y}{14}\right)+196y=1792$$

Express 8\times \frac{31+24y}{14} as a single fraction.

$$14\left(\frac{961+1488y+772y^{2}}{14^{2}}-\frac{8\left(31+24y\right)}{14}\right)+196y=1792$$

Use the distributive property to multiply 8 by 31+24y.

$$14\left(\frac{961+1488y+772y^{2}}{14^{2}}-\frac{248+192y}{14}\right)+196y=1792$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 14^{2} and 14 is 196. Multiply \frac{248+192y}{14} times \frac{14}{14}.

$$14\left(\frac{961+1488y+772y^{2}}{196}-\frac{14\left(248+192y\right)}{196}\right)+196y=1792$$

Since \frac{961+1488y+772y^{2}}{196} and \frac{14\left(248+192y\right)}{196} have the same denominator, subtract them by subtracting their numerators.

$$14\times \frac{961+1488y+772y^{2}-14\left(248+192y\right)}{196}+196y=1792$$

Do the multiplications in 961+1488y+772y^{2}-14\left(248+192y\right).

$$14\times \frac{961+1488y+772y^{2}-3472-2688y}{196}+196y=1792$$

Combine like terms in 961+1488y+772y^{2}-3472-2688y.

$$14\times \frac{-2511-1200y+772y^{2}}{196}+196y=1792$$

Cancel out 196, the greatest common factor in 14 and 196.

$$\frac{-2511-1200y+772y^{2}}{14}+196y=1792$$

Divide each term of -2511-1200y+772y^{2} by 14 to get -\frac{2511}{14}-\frac{600}{7}y+\frac{386}{7}y^{2}.

$$-\frac{2511}{14}-\frac{600}{7}y+\frac{386}{7}y^{2}+196y=1792$$

Combine -\frac{600}{7}y and 196y to get \frac{772}{7}y.

$$-\frac{2511}{14}+\frac{772}{7}y+\frac{386}{7}y^{2}=1792$$

Add \frac{2511}{14} to both sides.

$$\frac{772}{7}y+\frac{386}{7}y^{2}=1792+\frac{2511}{14}$$

Add 1792 and \frac{2511}{14} to get \frac{27599}{14}.

$$\frac{772}{7}y+\frac{386}{7}y^{2}=\frac{27599}{14}$$

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.

$$\frac{386}{7}y^{2}+\frac{772}{7}y=\frac{27599}{14}$$

Divide both sides of the equation by \frac{386}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{386}{7}y^{2}+\frac{772}{7}y}{\frac{386}{7}}=\frac{\frac{27599}{14}}{\frac{386}{7}}$$

Dividing by \frac{386}{7} undoes the multiplication by \frac{386}{7}.

$$y^{2}+\frac{\frac{772}{7}}{\frac{386}{7}}y=\frac{\frac{27599}{14}}{\frac{386}{7}}$$

Divide \frac{772}{7} by \frac{386}{7} by multiplying \frac{772}{7} by the reciprocal of \frac{386}{7}.

$$y^{2}+2y=\frac{\frac{27599}{14}}{\frac{386}{7}}$$

Divide \frac{27599}{14} by \frac{386}{7} by multiplying \frac{27599}{14} by the reciprocal of \frac{386}{7}.

$$y^{2}+2y=\frac{143}{4}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+2y+1^{2}=\frac{143}{4}+1^{2}$$

Square 1.

$$y^{2}+2y+1=\frac{143}{4}+1$$

Add \frac{143}{4} to 1.

$$y^{2}+2y+1=\frac{147}{4}$$

Factor y^{2}+2y+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+1\right)^{2}=\frac{147}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+1\right)^{2}}=\sqrt{\frac{147}{4}}$$

Simplify.

$$y+1=\frac{7\sqrt{3}}{2} y+1=-\frac{7\sqrt{3}}{2}$$

Subtract 1 from both sides of the equation.

$$y=\frac{7\sqrt{3}}{2}-1 y=-\frac{7\sqrt{3}}{2}-1$$