Simplify 3/2 times a plus a^2 minus 4/5 times a minus 3 times a^2.

Step-by-step solution

1. 1

   Sort the terms by degree before touching the fractions. There are two $a^2$ terms and two $a$ terms; they never mix, so handle each family separately:

   $\left(a^{2}-3a^{2}\right)+\left(\frac{3}{2}a-\frac{4}{5}a\right).$

2. 2

   Combine the quadratic terms. The first has an invisible coefficient of $1$:

   $1a^{2}-3a^{2}=-2a^{2}.$

3. 3

   Give the linear coefficients a common denominator. The denominators $2$ and $5$ have LCD $10$:

   $\frac{3}{2}=\frac{15}{10},\qquad \frac{4}{5}=\frac{8}{10}.$

4. 4

   Subtract the linear coefficients.

   $\frac{15}{10}-\frac{8}{10}=\frac{7}{10},$

   so the linear term is $\tfrac{7}{10}a$, or equivalently $0.7a$.

5. 5

   Write the simplified expression. Conventionally in descending powers:

   $-2a^{2}+\frac{7}{10}a.$

   It could also be factored as $a\left(\tfrac{7}{10}-2a\right)$, but as written it is already fully simplified.

6. 6

   Check with a value. At $a=10$ the original is $15+100-8-300=-193$, and the answer gives $-2(100)+\tfrac{7}{10}(10)=-200+7=-193$ ✓.