Solve 12left(50+left(12-1right)5right)/2+47=yleft(80+left(y-1right)5right)/2+37

$$\frac{ 12 \left( 50+ \left( 12-1 \right) 5 \right) }{ 2 } +47 = \frac{ y \left( 80+ \left( y-1 \right) 5 \right) }{ 2 } +37$$

Solve for y

$$y = \frac{\sqrt{1249} - 15}{2} \approx 10.170597047$$

$$y=\frac{-\sqrt{1249}-15}{2}\approx -25.170597047$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$12\left(50+\left(12-1\right)\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Subtract 1 from 12 to get 11.

$$12\left(50+11\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 11 and 5 to get 55.

$$12\left(50+55\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 50 and 55 to get 105.

$$12\times 105+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 12 and 105 to get 1260.

$$1260+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 1260 and 94 to get 1354.

$$1354=y\left(80+\left(y-1\right)\times 5\right)+74$$

Use the distributive property to multiply y-1 by 5.

$$1354=y\left(80+5y-5\right)+74$$

Subtract 5 from 80 to get 75.

$$1354=y\left(75+5y\right)+74$$

Use the distributive property to multiply y by 75+5y.

$$1354=75y+5y^{2}+74$$

Swap sides so that all variable terms are on the left hand side.

$$75y+5y^{2}+74=1354$$

Subtract 1354 from both sides.

$$75y+5y^{2}+74-1354=0$$

Subtract 1354 from 74 to get -1280.

$$75y+5y^{2}-1280=0$$

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.

$$5y^{2}+75y-1280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 75 for b, and -1280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-75±\sqrt{75^{2}-4\times 5\left(-1280\right)}}{2\times 5}$$

Square 75.

$$y=\frac{-75±\sqrt{5625-4\times 5\left(-1280\right)}}{2\times 5}$$

Multiply -4 times 5.

$$y=\frac{-75±\sqrt{5625-20\left(-1280\right)}}{2\times 5}$$

Multiply -20 times -1280.

$$y=\frac{-75±\sqrt{5625+25600}}{2\times 5}$$

Add 5625 to 25600.

$$y=\frac{-75±\sqrt{31225}}{2\times 5}$$

Take the square root of 31225.

$$y=\frac{-75±5\sqrt{1249}}{2\times 5}$$

Multiply 2 times 5.

$$y=\frac{-75±5\sqrt{1249}}{10}$$

Now solve the equation y=\frac{-75±5\sqrt{1249}}{10} when ± is plus. Add -75 to 5\sqrt{1249}.

$$y=\frac{5\sqrt{1249}-75}{10}$$

Divide -75+5\sqrt{1249} by 10.

$$y=\frac{\sqrt{1249}-15}{2}$$

Now solve the equation y=\frac{-75±5\sqrt{1249}}{10} when ± is minus. Subtract 5\sqrt{1249} from -75.

$$y=\frac{-5\sqrt{1249}-75}{10}$$

Divide -75-5\sqrt{1249} by 10.

$$y=\frac{-\sqrt{1249}-15}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{1249}-15}{2} y=\frac{-\sqrt{1249}-15}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$12\left(50+\left(12-1\right)\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Subtract 1 from 12 to get 11.

$$12\left(50+11\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 11 and 5 to get 55.

$$12\left(50+55\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 50 and 55 to get 105.

$$12\times 105+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 12 and 105 to get 1260.

$$1260+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 1260 and 94 to get 1354.

$$1354=y\left(80+\left(y-1\right)\times 5\right)+74$$

Use the distributive property to multiply y-1 by 5.

$$1354=y\left(80+5y-5\right)+74$$

Subtract 5 from 80 to get 75.

$$1354=y\left(75+5y\right)+74$$

Use the distributive property to multiply y by 75+5y.

$$1354=75y+5y^{2}+74$$

Swap sides so that all variable terms are on the left hand side.

$$75y+5y^{2}+74=1354$$

Subtract 74 from both sides.

$$75y+5y^{2}=1354-74$$

Subtract 74 from 1354 to get 1280.

$$75y+5y^{2}=1280$$

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.

$$5y^{2}+75y=1280$$

Divide both sides by 5.

$$\frac{5y^{2}+75y}{5}=\frac{1280}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$y^{2}+\frac{75}{5}y=\frac{1280}{5}$$

Divide 75 by 5.

$$y^{2}+15y=\frac{1280}{5}$$

Divide 1280 by 5.

$$y^{2}+15y=256$$

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

$$y^{2}+15y+\left(\frac{15}{2}\right)^{2}=256+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+15y+\frac{225}{4}=256+\frac{225}{4}$$

Add 256 to \frac{225}{4}.

$$y^{2}+15y+\frac{225}{4}=\frac{1249}{4}$$

Factor y^{2}+15y+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{15}{2}\right)^{2}=\frac{1249}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1249}{4}}$$

Simplify.

$$y+\frac{15}{2}=\frac{\sqrt{1249}}{2} y+\frac{15}{2}=-\frac{\sqrt{1249}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$y=\frac{\sqrt{1249}-15}{2} y=\frac{-\sqrt{1249}-15}{2}$$