Solve =a(-b/2a)^2+b(-b/2a)+c

$$= a ( - \frac { b } { 2 a } ) ^ { 2 } + b ( - \frac { b } { 2 a } ) + c$$

Evaluate

$$-\frac{b^{2}}{4a}+c$$

Short Solution Steps

Calculate -\frac{b}{2a} to the power of 2 and get \left(\frac{b}{2a}\right)^{2}.

$$a\times \left(\frac{b}{2a}\right)^{2}+b\left(-\frac{b}{2a}\right)+c$$

Express b\left(-\frac{b}{2a}\right) as a single fraction.

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-bb}{2a}+c$$

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

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-bb}{2a}+\frac{c\times 2a}{2a}$$

Since \frac{-bb}{2a} and \frac{c\times 2a}{2a} have the same denominator, add them by adding their numerators.

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-bb+c\times 2a}{2a}$$

Do the multiplications in -bb+c\times 2a.

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-b^{2}+c\times 2a}{2a}$$

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

$$a\times \frac{b^{2}}{\left(2a\right)^{2}}+\frac{-b^{2}+c\times 2a}{2a}$$

Express a\times \frac{b^{2}}{\left(2a\right)^{2}} as a single fraction.

$$\frac{ab^{2}}{\left(2a\right)^{2}}+\frac{-b^{2}+c\times 2a}{2a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2a\right)^{2} and 2a is 4a^{2}. Multiply \frac{-b^{2}+c\times 2a}{2a} times \frac{2a}{2a}.

$$\frac{ab^{2}}{4a^{2}}+\frac{\left(-b^{2}+c\times 2a\right)\times 2a}{4a^{2}}$$

Since \frac{ab^{2}}{4a^{2}} and \frac{\left(-b^{2}+c\times 2a\right)\times 2a}{4a^{2}} have the same denominator, add them by adding their numerators.

$$\frac{ab^{2}+\left(-b^{2}+c\times 2a\right)\times 2a}{4a^{2}}$$

Do the multiplications in ab^{2}+\left(-b^{2}+c\times 2a\right)\times 2a.

$$\frac{ab^{2}-2b^{2}a+4ca^{2}}{4a^{2}}$$

Combine like terms in ab^{2}-2b^{2}a+4ca^{2}.

$$\frac{-ab^{2}+4ca^{2}}{4a^{2}}$$

Factor the expressions that are not already factored in \frac{-ab^{2}+4ca^{2}}{4a^{2}}.

$$\frac{a\left(4ac-b^{2}\right)}{4a^{2}}$$

Cancel out a in both numerator and denominator.

$$\frac{4ac-b^{2}}{4a}$$

Expand

$$-\frac{b^{2}}{4a}+c$$

Short Solution Steps

Calculate -\frac{b}{2a} to the power of 2 and get \left(\frac{b}{2a}\right)^{2}.

$$a\times \left(\frac{b}{2a}\right)^{2}+b\left(-\frac{b}{2a}\right)+c$$

Express b\left(-\frac{b}{2a}\right) as a single fraction.

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-bb}{2a}+c$$

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

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-bb}{2a}+\frac{c\times 2a}{2a}$$

Since \frac{-bb}{2a} and \frac{c\times 2a}{2a} have the same denominator, add them by adding their numerators.

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-bb+c\times 2a}{2a}$$

Do the multiplications in -bb+c\times 2a.

$$a\times \left(\frac{b}{2a}\right)^{2}+\frac{-b^{2}+c\times 2a}{2a}$$

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

$$a\times \frac{b^{2}}{\left(2a\right)^{2}}+\frac{-b^{2}+c\times 2a}{2a}$$

Express a\times \frac{b^{2}}{\left(2a\right)^{2}} as a single fraction.

$$\frac{ab^{2}}{\left(2a\right)^{2}}+\frac{-b^{2}+c\times 2a}{2a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2a\right)^{2} and 2a is 4a^{2}. Multiply \frac{-b^{2}+c\times 2a}{2a} times \frac{2a}{2a}.

$$\frac{ab^{2}}{4a^{2}}+\frac{\left(-b^{2}+c\times 2a\right)\times 2a}{4a^{2}}$$

Since \frac{ab^{2}}{4a^{2}} and \frac{\left(-b^{2}+c\times 2a\right)\times 2a}{4a^{2}} have the same denominator, add them by adding their numerators.

$$\frac{ab^{2}+\left(-b^{2}+c\times 2a\right)\times 2a}{4a^{2}}$$

Do the multiplications in ab^{2}+\left(-b^{2}+c\times 2a\right)\times 2a.

$$\frac{ab^{2}-2b^{2}a+4ca^{2}}{4a^{2}}$$

Combine like terms in ab^{2}-2b^{2}a+4ca^{2}.

$$\frac{-ab^{2}+4ca^{2}}{4a^{2}}$$

Factor the expressions that are not already factored in \frac{-ab^{2}+4ca^{2}}{4a^{2}}.

$$\frac{a\left(4ac-b^{2}\right)}{4a^{2}}$$

Cancel out a in both numerator and denominator.

$$\frac{4ac-b^{2}}{4a}$$