Solve S=4(w/2)^2+35/2

$$S = 4 ( \frac { w } { 2 } ) ^ { 2 } + \frac { 35 } { 2 }$$

Solve for S

$$S=w^{2}+\frac{35}{2}$$

Short Solution Steps

To raise \frac{w}{2} to a power, raise both numerator and denominator to the power and then divide.

$$S=4\times \frac{w^{2}}{2^{2}}+\frac{35}{2}$$

Express 4\times \frac{w^{2}}{2^{2}} as a single fraction.

$$S=\frac{4w^{2}}{2^{2}}+\frac{35}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 2 is 4. Multiply \frac{35}{2} times \frac{2}{2}.

$$S=\frac{4w^{2}}{4}+\frac{35\times 2}{4}$$

Since \frac{4w^{2}}{4} and \frac{35\times 2}{4} have the same denominator, add them by adding their numerators.

$$S=\frac{4w^{2}+35\times 2}{4}$$

Do the multiplications in 4w^{2}+35\times 2.

$$S=\frac{4w^{2}+70}{4}$$

Divide each term of 4w^{2}+70 by 4 to get w^{2}+\frac{35}{2}.

$$S=w^{2}+\frac{35}{2}$$

Solve for w

$$w=\frac{\sqrt{4S-70}}{2}$$

$$w=-\frac{\sqrt{4S-70}}{2}\text{, }S\geq \frac{35}{2}$$