Solve (1/2-a)^2-3(a-1/2)(a+1/2)+2(a-1)^2

$$( \frac { 1 } { 2 } - a ) ^ { 2 } - 3 ( a - \frac { 1 } { 2 } ) ( a + \frac { 1 } { 2 } ) + 2 ( a - 1 ) ^ { 2 }$$

Evaluate

$$3-5a$$

Solution Steps

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

$$\frac{1}{4}-a+a^{2}-3\left(a-\frac{1}{2}\right)\left(a+\frac{1}{2}\right)+2\left(a-1\right)^{2}$$

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

$$\frac{1}{4}-a+a^{2}-3\left(a-\frac{1}{2}\right)\left(a+\frac{1}{2}\right)+2\left(a^{2}-2a+1\right)$$

Use the distributive property to multiply 2 by a^{2}-2a+1.

$$\frac{1}{4}-a+a^{2}-3\left(a-\frac{1}{2}\right)\left(a+\frac{1}{2}\right)+2a^{2}-4a+2$$

Use the distributive property to multiply -3 by a-\frac{1}{2}.

$$\frac{1}{4}-a+a^{2}+\left(-3a+\frac{3}{2}\right)\left(a+\frac{1}{2}\right)+2a^{2}-4a+2$$

Use the distributive property to multiply -3a+\frac{3}{2} by a+\frac{1}{2} and combine like terms.

$$\frac{1}{4}-a+a^{2}-3a^{2}+\frac{3}{4}+2a^{2}-4a+2$$

Combine a^{2} and -3a^{2} to get -2a^{2}.

$$\frac{1}{4}-a-2a^{2}+\frac{3}{4}+2a^{2}-4a+2$$

Add \frac{1}{4} and \frac{3}{4} to get 1.

$$1-a-2a^{2}+2a^{2}-4a+2$$

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

$$1-a-4a+2$$

Combine -a and -4a to get -5a.

$$1-5a+2$$

Add 1 and 2 to get 3.

$$3-5a$$

Expand

$$3-5a$$

Solution Steps

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

$$\frac{1}{4}-a+a^{2}-3\left(a-\frac{1}{2}\right)\left(a+\frac{1}{2}\right)+2\left(a-1\right)^{2}$$

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

$$\frac{1}{4}-a+a^{2}-3\left(a-\frac{1}{2}\right)\left(a+\frac{1}{2}\right)+2\left(a^{2}-2a+1\right)$$

Use the distributive property to multiply 2 by a^{2}-2a+1.

$$\frac{1}{4}-a+a^{2}-3\left(a-\frac{1}{2}\right)\left(a+\frac{1}{2}\right)+2a^{2}-4a+2$$

Use the distributive property to multiply -3 by a-\frac{1}{2}.

$$\frac{1}{4}-a+a^{2}+\left(-3a+\frac{3}{2}\right)\left(a+\frac{1}{2}\right)+2a^{2}-4a+2$$

Use the distributive property to multiply -3a+\frac{3}{2} by a+\frac{1}{2} and combine like terms.

$$\frac{1}{4}-a+a^{2}-3a^{2}+\frac{3}{4}+2a^{2}-4a+2$$

Combine a^{2} and -3a^{2} to get -2a^{2}.

$$\frac{1}{4}-a-2a^{2}+\frac{3}{4}+2a^{2}-4a+2$$

Add \frac{1}{4} and \frac{3}{4} to get 1.

$$1-a-2a^{2}+2a^{2}-4a+2$$

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

$$1-a-4a+2$$

Combine -a and -4a to get -5a.

$$1-5a+2$$

Add 1 and 2 to get 3.

$$3-5a$$