Solve (19-3y/2)^2+y^2-10(14-3y/2)-6y+86=0

$$( \frac { 19 - 3 y } { 2 } ) ^ { 2 } + y ^ { 2 } - 10 ( \frac { 14 - 3 y } { 2 } ) - 6 y + 86 = 0$$

Solve for y

$$y=\frac{2\sqrt{1001}i}{13}+3\approx 3+4.867474468i$$

$$y=-\frac{2\sqrt{1001}i}{13}+3\approx 3-4.867474468i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\left(\left(\frac{19-3y}{2}\right)^{2}+y^{2}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

To raise \frac{19-3y}{2} to a power, raise both numerator and denominator to the power and then divide.

$$2\left(\frac{\left(19-3y\right)^{2}}{2^{2}}+y^{2}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

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

$$2\left(\frac{\left(19-3y\right)^{2}}{2^{2}}+\frac{y^{2}\times 2^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Since \frac{\left(19-3y\right)^{2}}{2^{2}} and \frac{y^{2}\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$2\left(\frac{\left(19-3y\right)^{2}+y^{2}\times 2^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Do the multiplications in \left(19-3y\right)^{2}+y^{2}\times 2^{2}.

$$2\left(\frac{361-114y+9y^{2}+4y^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Combine like terms in 361-114y+9y^{2}+4y^{2}.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Cancel out 2, the greatest common factor in 10 and 2.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}-5\left(14-3y\right)\right)-12y+172=0$$

Use the distributive property to multiply -5 by 14-3y.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}-70+15y\right)-12y+172=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -70+15y times \frac{2^{2}}{2^{2}}.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}+\frac{\left(-70+15y\right)\times 2^{2}}{2^{2}}\right)-12y+172=0$$

Since \frac{361-114y+13y^{2}}{2^{2}} and \frac{\left(-70+15y\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$2\times \frac{361-114y+13y^{2}+\left(-70+15y\right)\times 2^{2}}{2^{2}}-12y+172=0$$

Do the multiplications in 361-114y+13y^{2}+\left(-70+15y\right)\times 2^{2}.

$$2\times \frac{361-114y+13y^{2}-280+60y}{2^{2}}-12y+172=0$$

Combine like terms in 361-114y+13y^{2}-280+60y.

$$2\times \frac{81-54y+13y^{2}}{2^{2}}-12y+172=0$$

Express 2\times \frac{81-54y+13y^{2}}{2^{2}} as a single fraction.

$$\frac{2\left(81-54y+13y^{2}\right)}{2^{2}}-12y+172=0$$

Cancel out 2 in both numerator and denominator.

$$\frac{13y^{2}-54y+81}{2}-12y+172=0$$

Divide each term of 13y^{2}-54y+81 by 2 to get \frac{13}{2}y^{2}-27y+\frac{81}{2}.

$$\frac{13}{2}y^{2}-27y+\frac{81}{2}-12y+172=0$$

Combine -27y and -12y to get -39y.

$$\frac{13}{2}y^{2}-39y+\frac{81}{2}+172=0$$

Add \frac{81}{2} and 172 to get \frac{425}{2}.

$$\frac{13}{2}y^{2}-39y+\frac{425}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{13}{2} for a, -39 for b, and \frac{425}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-39\right)±\sqrt{\left(-39\right)^{2}-4\times \frac{13}{2}\times \frac{425}{2}}}{2\times \frac{13}{2}}$$

Square -39.

$$y=\frac{-\left(-39\right)±\sqrt{1521-4\times \frac{13}{2}\times \frac{425}{2}}}{2\times \frac{13}{2}}$$

Multiply -4 times \frac{13}{2}.

$$y=\frac{-\left(-39\right)±\sqrt{1521-26\times \frac{425}{2}}}{2\times \frac{13}{2}}$$

Multiply -26 times \frac{425}{2}.

$$y=\frac{-\left(-39\right)±\sqrt{1521-5525}}{2\times \frac{13}{2}}$$

Add 1521 to -5525.

$$y=\frac{-\left(-39\right)±\sqrt{-4004}}{2\times \frac{13}{2}}$$

Take the square root of -4004.

$$y=\frac{-\left(-39\right)±2\sqrt{1001}i}{2\times \frac{13}{2}}$$

The opposite of -39 is 39.

$$y=\frac{39±2\sqrt{1001}i}{2\times \frac{13}{2}}$$

Multiply 2 times \frac{13}{2}.

$$y=\frac{39±2\sqrt{1001}i}{13}$$

Now solve the equation y=\frac{39±2\sqrt{1001}i}{13} when ± is plus. Add 39 to 2i\sqrt{1001}.

$$y=\frac{39+2\sqrt{1001}i}{13}$$

Divide 39+2i\sqrt{1001} by 13.

$$y=\frac{2\sqrt{1001}i}{13}+3$$

Now solve the equation y=\frac{39±2\sqrt{1001}i}{13} when ± is minus. Subtract 2i\sqrt{1001} from 39.

$$y=\frac{-2\sqrt{1001}i+39}{13}$$

Divide 39-2i\sqrt{1001} by 13.

$$y=-\frac{2\sqrt{1001}i}{13}+3$$

The equation is now solved.

$$y=\frac{2\sqrt{1001}i}{13}+3 y=-\frac{2\sqrt{1001}i}{13}+3$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\left(\left(\frac{19-3y}{2}\right)^{2}+y^{2}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

To raise \frac{19-3y}{2} to a power, raise both numerator and denominator to the power and then divide.

$$2\left(\frac{\left(19-3y\right)^{2}}{2^{2}}+y^{2}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

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

$$2\left(\frac{\left(19-3y\right)^{2}}{2^{2}}+\frac{y^{2}\times 2^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Since \frac{\left(19-3y\right)^{2}}{2^{2}} and \frac{y^{2}\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$2\left(\frac{\left(19-3y\right)^{2}+y^{2}\times 2^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Do the multiplications in \left(19-3y\right)^{2}+y^{2}\times 2^{2}.

$$2\left(\frac{361-114y+9y^{2}+4y^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Combine like terms in 361-114y+9y^{2}+4y^{2}.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}-10\times \frac{14-3y}{2}\right)-12y+172=0$$

Cancel out 2, the greatest common factor in 10 and 2.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}-5\left(14-3y\right)\right)-12y+172=0$$

Use the distributive property to multiply -5 by 14-3y.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}-70+15y\right)-12y+172=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -70+15y times \frac{2^{2}}{2^{2}}.

$$2\left(\frac{361-114y+13y^{2}}{2^{2}}+\frac{\left(-70+15y\right)\times 2^{2}}{2^{2}}\right)-12y+172=0$$

Since \frac{361-114y+13y^{2}}{2^{2}} and \frac{\left(-70+15y\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$2\times \frac{361-114y+13y^{2}+\left(-70+15y\right)\times 2^{2}}{2^{2}}-12y+172=0$$

Do the multiplications in 361-114y+13y^{2}+\left(-70+15y\right)\times 2^{2}.

$$2\times \frac{361-114y+13y^{2}-280+60y}{2^{2}}-12y+172=0$$

Combine like terms in 361-114y+13y^{2}-280+60y.

$$2\times \frac{81-54y+13y^{2}}{2^{2}}-12y+172=0$$

Express 2\times \frac{81-54y+13y^{2}}{2^{2}} as a single fraction.

$$\frac{2\left(81-54y+13y^{2}\right)}{2^{2}}-12y+172=0$$

Cancel out 2 in both numerator and denominator.

$$\frac{13y^{2}-54y+81}{2}-12y+172=0$$

Divide each term of 13y^{2}-54y+81 by 2 to get \frac{13}{2}y^{2}-27y+\frac{81}{2}.

$$\frac{13}{2}y^{2}-27y+\frac{81}{2}-12y+172=0$$

Combine -27y and -12y to get -39y.

$$\frac{13}{2}y^{2}-39y+\frac{81}{2}+172=0$$

Add \frac{81}{2} and 172 to get \frac{425}{2}.

$$\frac{13}{2}y^{2}-39y+\frac{425}{2}=0$$

Subtract \frac{425}{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{13}{2}y^{2}-39y=-\frac{425}{2}$$

Divide both sides of the equation by \frac{13}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{13}{2}y^{2}-39y}{\frac{13}{2}}=-\frac{\frac{425}{2}}{\frac{13}{2}}$$

Dividing by \frac{13}{2} undoes the multiplication by \frac{13}{2}.

$$y^{2}+\left(-\frac{39}{\frac{13}{2}}\right)y=-\frac{\frac{425}{2}}{\frac{13}{2}}$$

Divide -39 by \frac{13}{2} by multiplying -39 by the reciprocal of \frac{13}{2}.

$$y^{2}-6y=-\frac{\frac{425}{2}}{\frac{13}{2}}$$

Divide -\frac{425}{2} by \frac{13}{2} by multiplying -\frac{425}{2} by the reciprocal of \frac{13}{2}.

$$y^{2}-6y=-\frac{425}{13}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-6y+\left(-3\right)^{2}=-\frac{425}{13}+\left(-3\right)^{2}$$

Square -3.

$$y^{2}-6y+9=-\frac{425}{13}+9$$

Add -\frac{425}{13} to 9.

$$y^{2}-6y+9=-\frac{308}{13}$$

Factor y^{2}-6y+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-3\right)^{2}=-\frac{308}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-3\right)^{2}}=\sqrt{-\frac{308}{13}}$$

Simplify.

$$y-3=\frac{2\sqrt{1001}i}{13} y-3=-\frac{2\sqrt{1001}i}{13}$$

Add 3 to both sides of the equation.

$$y=\frac{2\sqrt{1001}i}{13}+3 y=-\frac{2\sqrt{1001}i}{13}+3$$