Solve M=(-b+1/2a)^2-(b-b(a-3))-(ab^3-0.75a^3b):(ab)

$$M = ( - b + \frac { 1 } { 2 } a ) ^ { 2 } - ( b - b ( a - 3 ) ) - ( a b ^ { 3 } - 0.75 a ^ { 3 } b ) : ( a b )$$

Solve for M

$$M=a^{2}-4b$$

$$a\neq 0\text{ and }b\neq 0$$

Short Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(-b+\frac{1}{2}a\right)^{2}.

$$M=\left(-b\right)^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-\left(b-b\left(a-3\right)\right)-\frac{ab^{3}-0.75a^{3}b}{ab}$$

Calculate -b to the power of 2 and get b^{2}.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-\left(b-b\left(a-3\right)\right)-\frac{ab^{3}-0.75a^{3}b}{ab}$$

Use the distributive property to multiply b by a-3.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-\left(b-\left(ba-3b\right)\right)-\frac{ab^{3}-0.75a^{3}b}{ab}$$

To find the opposite of ba-3b, find the opposite of each term.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-\left(b-ba+3b\right)-\frac{ab^{3}-0.75a^{3}b}{ab}$$

Combine b and 3b to get 4b.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-\left(4b-ba\right)-\frac{ab^{3}-0.75a^{3}b}{ab}$$

To find the opposite of 4b-ba, find the opposite of each term.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-4b+ba-\frac{ab^{3}-0.75a^{3}b}{ab}$$

Factor the expressions that are not already factored in \frac{ab^{3}-0.75a^{3}b}{ab}.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-4b+ba-\frac{0.25ab\left(-3a^{2}+4b^{2}\right)}{ab}$$

Cancel out ab in both numerator and denominator.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-4b+ba-0.25\left(-3a^{2}+4b^{2}\right)$$

Expand the expression.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-4b+ba-\left(-0.75a^{2}+b^{2}\right)$$

To find the opposite of -0.75a^{2}+b^{2}, find the opposite of each term.

$$M=b^{2}+\left(-b\right)a+\frac{1}{4}a^{2}-4b+ba+0.75a^{2}-b^{2}$$

Combine \frac{1}{4}a^{2} and 0.75a^{2} to get a^{2}.

$$M=b^{2}+\left(-b\right)a+a^{2}-4b+ba-b^{2}$$

Combine b^{2} and -b^{2} to get 0.

$$M=\left(-b\right)a+a^{2}-4b+ba$$

Combine -ba and ba to get 0.

$$M=a^{2}-4b$$

Solve for a

$$a=\sqrt{M+4b}$$

$$a=-\sqrt{M+4b}\text{, }M>-4b\text{ and }b\neq 0$$