Solve =(3/2-2x)(3/2+2x)-(2x+1/2)

$$= ( \frac { 3 } { 2 } - 2 x ) ( \frac { 3 } { 2 } + 2 x ) - ( 2 x + \frac { 1 } { 2 } )$$

Evaluate

$$\frac{7}{4}-2x-4x^{2}$$

Solution Steps

Consider \left(\frac{3}{2}-2x\right)\left(\frac{3}{2}+2x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{9}{4}-\left(2x\right)^{2}-\left(2x+\frac{1}{2}\right)$$

Expand \left(2x\right)^{2}.

$$\frac{9}{4}-2^{2}x^{2}-\left(2x+\frac{1}{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{9}{4}-4x^{2}-\left(2x+\frac{1}{2}\right)$$

To find the opposite of 2x+\frac{1}{2}, find the opposite of each term.

$$\frac{9}{4}-4x^{2}-2x-\frac{1}{2}$$

Least common multiple of 4 and 2 is 4. Convert \frac{9}{4} and \frac{1}{2} to fractions with denominator 4.

$$\frac{9}{4}-4x^{2}-2x-\frac{2}{4}$$

Since \frac{9}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9-2}{4}-4x^{2}-2x$$

Subtract 2 from 9 to get 7.

$$\frac{7}{4}-4x^{2}-2x$$

Expand

$$\frac{7}{4}-2x-4x^{2}$$

Solution Steps

Consider \left(\frac{3}{2}-2x\right)\left(\frac{3}{2}+2x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{9}{4}-\left(2x\right)^{2}-\left(2x+\frac{1}{2}\right)$$

Expand \left(2x\right)^{2}.

$$\frac{9}{4}-2^{2}x^{2}-\left(2x+\frac{1}{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{9}{4}-4x^{2}-\left(2x+\frac{1}{2}\right)$$

To find the opposite of 2x+\frac{1}{2}, find the opposite of each term.

$$\frac{9}{4}-4x^{2}-2x-\frac{1}{2}$$

Least common multiple of 4 and 2 is 4. Convert \frac{9}{4} and \frac{1}{2} to fractions with denominator 4.

$$\frac{9}{4}-4x^{2}-2x-\frac{2}{4}$$

Since \frac{9}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9-2}{4}-4x^{2}-2x$$

Subtract 2 from 9 to get 7.

$$\frac{7}{4}-4x^{2}-2x$$