Solve y={left(y+12/4right)}^2+2(y+12/4)+13

$$y= { \left( \frac{ y+12 }{ 4 } \right) }^{ 2 } +2( \frac{ y+12 }{ 4 } )+13$$

Solve for y

$$y=-8+8\sqrt{6}i\approx -8+19.595917942i$$

$$y=-8\sqrt{6}i-8\approx -8-19.595917942i$$

Steps Using the Quadratic Formula

To raise \frac{y+12}{4} to a power, raise both numerator and denominator to the power and then divide.

$$y=\frac{\left(y+12\right)^{2}}{4^{2}}+2\times \frac{y+12}{4}+13$$

Cancel out 4, the greatest common factor in 2 and 4.

$$y=\frac{\left(y+12\right)^{2}}{4^{2}}+\frac{y+12}{2}+13$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 2 is 16. Multiply \frac{y+12}{2} times \frac{8}{8}.

$$y=\frac{\left(y+12\right)^{2}}{16}+\frac{8\left(y+12\right)}{16}+13$$

Since \frac{\left(y+12\right)^{2}}{16} and \frac{8\left(y+12\right)}{16} have the same denominator, add them by adding their numerators.

$$y=\frac{\left(y+12\right)^{2}+8\left(y+12\right)}{16}+13$$

Do the multiplications in \left(y+12\right)^{2}+8\left(y+12\right).

$$y=\frac{y^{2}+24y+144+8y+96}{16}+13$$

Combine like terms in y^{2}+24y+144+8y+96.

$$y=\frac{y^{2}+32y+240}{16}+13$$

Divide each term of y^{2}+32y+240 by 16 to get \frac{1}{16}y^{2}+2y+15.

$$y=\frac{1}{16}y^{2}+2y+15+13$$

Add 15 and 13 to get 28.

$$y=\frac{1}{16}y^{2}+2y+28$$

Subtract \frac{1}{16}y^{2} from both sides.

$$y-\frac{1}{16}y^{2}=2y+28$$

Subtract 2y from both sides.

$$y-\frac{1}{16}y^{2}-2y=28$$

Combine y and -2y to get -y.

$$-y-\frac{1}{16}y^{2}=28$$

Subtract 28 from both sides.

$$-y-\frac{1}{16}y^{2}-28=0$$

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.

$$-\frac{1}{16}y^{2}-y-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{16} for a, -1 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{1}{16}\right)\left(-28\right)}}{2\left(-\frac{1}{16}\right)}$$

Multiply -4 times -\frac{1}{16}.

$$y=\frac{-\left(-1\right)±\sqrt{1+\frac{1}{4}\left(-28\right)}}{2\left(-\frac{1}{16}\right)}$$

Multiply \frac{1}{4} times -28.

$$y=\frac{-\left(-1\right)±\sqrt{1-7}}{2\left(-\frac{1}{16}\right)}$$

Add 1 to -7.

$$y=\frac{-\left(-1\right)±\sqrt{-6}}{2\left(-\frac{1}{16}\right)}$$

Take the square root of -6.

$$y=\frac{-\left(-1\right)±\sqrt{6}i}{2\left(-\frac{1}{16}\right)}$$

The opposite of -1 is 1.

$$y=\frac{1±\sqrt{6}i}{2\left(-\frac{1}{16}\right)}$$

Multiply 2 times -\frac{1}{16}.

$$y=\frac{1±\sqrt{6}i}{-\frac{1}{8}}$$

Now solve the equation y=\frac{1±\sqrt{6}i}{-\frac{1}{8}} when ± is plus. Add 1 to i\sqrt{6}.

$$y=\frac{1+\sqrt{6}i}{-\frac{1}{8}}$$

Divide 1+i\sqrt{6} by -\frac{1}{8} by multiplying 1+i\sqrt{6} by the reciprocal of -\frac{1}{8}.

$$y=-8\sqrt{6}i-8$$

Now solve the equation y=\frac{1±\sqrt{6}i}{-\frac{1}{8}} when ± is minus. Subtract i\sqrt{6} from 1.

$$y=\frac{-\sqrt{6}i+1}{-\frac{1}{8}}$$

Divide 1-i\sqrt{6} by -\frac{1}{8} by multiplying 1-i\sqrt{6} by the reciprocal of -\frac{1}{8}.

$$y=-8+8\sqrt{6}i$$

The equation is now solved.

$$y=-8\sqrt{6}i-8 y=-8+8\sqrt{6}i$$

Steps for Completing the Square

To raise \frac{y+12}{4} to a power, raise both numerator and denominator to the power and then divide.

$$y=\frac{\left(y+12\right)^{2}}{4^{2}}+2\times \frac{y+12}{4}+13$$

Cancel out 4, the greatest common factor in 2 and 4.

$$y=\frac{\left(y+12\right)^{2}}{4^{2}}+\frac{y+12}{2}+13$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4^{2} and 2 is 16. Multiply \frac{y+12}{2} times \frac{8}{8}.

$$y=\frac{\left(y+12\right)^{2}}{16}+\frac{8\left(y+12\right)}{16}+13$$

Since \frac{\left(y+12\right)^{2}}{16} and \frac{8\left(y+12\right)}{16} have the same denominator, add them by adding their numerators.

$$y=\frac{\left(y+12\right)^{2}+8\left(y+12\right)}{16}+13$$

Do the multiplications in \left(y+12\right)^{2}+8\left(y+12\right).

$$y=\frac{y^{2}+24y+144+8y+96}{16}+13$$

Combine like terms in y^{2}+24y+144+8y+96.

$$y=\frac{y^{2}+32y+240}{16}+13$$

Divide each term of y^{2}+32y+240 by 16 to get \frac{1}{16}y^{2}+2y+15.

$$y=\frac{1}{16}y^{2}+2y+15+13$$

Add 15 and 13 to get 28.

$$y=\frac{1}{16}y^{2}+2y+28$$

Subtract \frac{1}{16}y^{2} from both sides.

$$y-\frac{1}{16}y^{2}=2y+28$$

Subtract 2y from both sides.

$$y-\frac{1}{16}y^{2}-2y=28$$

Combine y and -2y to get -y.

$$-y-\frac{1}{16}y^{2}=28$$

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.

$$-\frac{1}{16}y^{2}-y=28$$

Multiply both sides by -16.

$$\frac{-\frac{1}{16}y^{2}-y}{-\frac{1}{16}}=\frac{28}{-\frac{1}{16}}$$

Dividing by -\frac{1}{16} undoes the multiplication by -\frac{1}{16}.

$$y^{2}+\left(-\frac{1}{-\frac{1}{16}}\right)y=\frac{28}{-\frac{1}{16}}$$

Divide -1 by -\frac{1}{16} by multiplying -1 by the reciprocal of -\frac{1}{16}.

$$y^{2}+16y=\frac{28}{-\frac{1}{16}}$$

Divide 28 by -\frac{1}{16} by multiplying 28 by the reciprocal of -\frac{1}{16}.

$$y^{2}+16y=-448$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+16y+8^{2}=-448+8^{2}$$

Square 8.

$$y^{2}+16y+64=-448+64$$

Add -448 to 64.

$$y^{2}+16y+64=-384$$

Factor y^{2}+16y+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}.

$$\left(y+8\right)^{2}=-384$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+8\right)^{2}}=\sqrt{-384}$$

Simplify.

$$y+8=8\sqrt{6}i y+8=-8\sqrt{6}i$$

Subtract 8 from both sides of the equation.

$$y=-8+8\sqrt{6}i y=-8\sqrt{6}i-8$$