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

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

Evaluate

$$\frac{15a^{2}}{2}-8a+\frac{13}{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by 4a^{2}-4a+1.

$$8a^{2}-8a+2-\frac{1}{2}\left(a+3\right)\left(a-3\right)$$

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

$$8a^{2}-8a+2+\left(-\frac{1}{2}a-\frac{3}{2}\right)\left(a-3\right)$$

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

$$8a^{2}-8a+2-\frac{1}{2}a^{2}+\frac{9}{2}$$

Combine 8a^{2} and -\frac{1}{2}a^{2} to get \frac{15}{2}a^{2}.

$$\frac{15}{2}a^{2}-8a+2+\frac{9}{2}$$

Add 2 and \frac{9}{2} to get \frac{13}{2}.

$$\frac{15}{2}a^{2}-8a+\frac{13}{2}$$

Expand

$$\frac{15a^{2}}{2}-8a+\frac{13}{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by 4a^{2}-4a+1.

$$8a^{2}-8a+2-\frac{1}{2}\left(a+3\right)\left(a-3\right)$$

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

$$8a^{2}-8a+2+\left(-\frac{1}{2}a-\frac{3}{2}\right)\left(a-3\right)$$

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

$$8a^{2}-8a+2-\frac{1}{2}a^{2}+\frac{9}{2}$$

Combine 8a^{2} and -\frac{1}{2}a^{2} to get \frac{15}{2}a^{2}.

$$\frac{15}{2}a^{2}-8a+2+\frac{9}{2}$$

Add 2 and \frac{9}{2} to get \frac{13}{2}.

$$\frac{15}{2}a^{2}-8a+\frac{13}{2}$$