Solve 4left(50+left(4-1right)5right)/2+47=yleft(80+left(y-1right)5right)/2+37

$$\frac{ 4 \left( 50+ \left( 4-1 \right) 5 \right) }{ 2 } +47 = \frac{ y \left( 80+ \left( y-1 \right) 5 \right) }{ 2 } +37$$

Solve for y

$$y = \frac{\sqrt{449} - 15}{2} \approx 3.09481005$$

$$y=\frac{-\sqrt{449}-15}{2}\approx -18.09481005$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$4\left(50+\left(4-1\right)\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Subtract 1 from 4 to get 3.

$$4\left(50+3\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 3 and 5 to get 15.

$$4\left(50+15\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 50 and 15 to get 65.

$$4\times 65+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 4 and 65 to get 260.

$$260+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 260 and 94 to get 354.

$$354=y\left(80+\left(y-1\right)\times 5\right)+74$$

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

$$354=y\left(80+5y-5\right)+74$$

Subtract 5 from 80 to get 75.

$$354=y\left(75+5y\right)+74$$

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

$$354=75y+5y^{2}+74$$

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

$$75y+5y^{2}+74=354$$

Subtract 354 from both sides.

$$75y+5y^{2}+74-354=0$$

Subtract 354 from 74 to get -280.

$$75y+5y^{2}-280=0$$

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.

$$5y^{2}+75y-280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 75 for b, and -280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-75±\sqrt{75^{2}-4\times 5\left(-280\right)}}{2\times 5}$$

Square 75.

$$y=\frac{-75±\sqrt{5625-4\times 5\left(-280\right)}}{2\times 5}$$

Multiply -4 times 5.

$$y=\frac{-75±\sqrt{5625-20\left(-280\right)}}{2\times 5}$$

Multiply -20 times -280.

$$y=\frac{-75±\sqrt{5625+5600}}{2\times 5}$$

Add 5625 to 5600.

$$y=\frac{-75±\sqrt{11225}}{2\times 5}$$

Take the square root of 11225.

$$y=\frac{-75±5\sqrt{449}}{2\times 5}$$

Multiply 2 times 5.

$$y=\frac{-75±5\sqrt{449}}{10}$$

Now solve the equation y=\frac{-75±5\sqrt{449}}{10} when ± is plus. Add -75 to 5\sqrt{449}.

$$y=\frac{5\sqrt{449}-75}{10}$$

Divide -75+5\sqrt{449} by 10.

$$y=\frac{\sqrt{449}-15}{2}$$

Now solve the equation y=\frac{-75±5\sqrt{449}}{10} when ± is minus. Subtract 5\sqrt{449} from -75.

$$y=\frac{-5\sqrt{449}-75}{10}$$

Divide -75-5\sqrt{449} by 10.

$$y=\frac{-\sqrt{449}-15}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{449}-15}{2} y=\frac{-\sqrt{449}-15}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$4\left(50+\left(4-1\right)\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Subtract 1 from 4 to get 3.

$$4\left(50+3\times 5\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 3 and 5 to get 15.

$$4\left(50+15\right)+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 50 and 15 to get 65.

$$4\times 65+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Multiply 4 and 65 to get 260.

$$260+94=y\left(80+\left(y-1\right)\times 5\right)+74$$

Add 260 and 94 to get 354.

$$354=y\left(80+\left(y-1\right)\times 5\right)+74$$

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

$$354=y\left(80+5y-5\right)+74$$

Subtract 5 from 80 to get 75.

$$354=y\left(75+5y\right)+74$$

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

$$354=75y+5y^{2}+74$$

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

$$75y+5y^{2}+74=354$$

Subtract 74 from both sides.

$$75y+5y^{2}=354-74$$

Subtract 74 from 354 to get 280.

$$75y+5y^{2}=280$$

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.

$$5y^{2}+75y=280$$

Divide both sides by 5.

$$\frac{5y^{2}+75y}{5}=\frac{280}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$y^{2}+\frac{75}{5}y=\frac{280}{5}$$

Divide 75 by 5.

$$y^{2}+15y=\frac{280}{5}$$

Divide 280 by 5.

$$y^{2}+15y=56$$

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.

$$y^{2}+15y+\left(\frac{15}{2}\right)^{2}=56+\left(\frac{15}{2}\right)^{2}$$

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

$$y^{2}+15y+\frac{225}{4}=56+\frac{225}{4}$$

Add 56 to \frac{225}{4}.

$$y^{2}+15y+\frac{225}{4}=\frac{449}{4}$$

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}.

$$\left(y+\frac{15}{2}\right)^{2}=\frac{449}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{15}{2}\right)^{2}}=\sqrt{\frac{449}{4}}$$

Simplify.

$$y+\frac{15}{2}=\frac{\sqrt{449}}{2} y+\frac{15}{2}=-\frac{\sqrt{449}}{2}$$

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

$$y=\frac{\sqrt{449}-15}{2} y=\frac{-\sqrt{449}-15}{2}$$