Solve x*x+2/3-(x-2)(x+2)/2-(x-2)^2-4

$$x \cdot \frac { x + 2 } { 3 } - \frac { ( x - 2 ) ( x + 2 ) } { 2 } - ( x - 2 ) ^ { 2 } - 4$$

Evaluate

$$-\frac{7x^{2}}{6}+\frac{14x}{3}-6$$

Short Solution Steps

Express x\times \frac{x+2}{3} as a single fraction.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x\left(x+2\right)}{3} times \frac{2}{2}. Multiply \frac{x^{2}-4}{2} times \frac{3}{3}.

$$\frac{2x\left(x+2\right)}{6}-\frac{3\left(x^{2}-4\right)}{6}-\left(x-2\right)^{2}-4$$

Since \frac{2x\left(x+2\right)}{6} and \frac{3\left(x^{2}-4\right)}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x\left(x+2\right)-3\left(x^{2}-4\right)}{6}-\left(x-2\right)^{2}-4$$

Do the multiplications in 2x\left(x+2\right)-3\left(x^{2}-4\right).

$$\frac{2x^{2}+4x-3x^{2}+12}{6}-\left(x-2\right)^{2}-4$$

Combine like terms in 2x^{2}+4x-3x^{2}+12.

$$\frac{-x^{2}+4x+12}{6}-\left(x-2\right)^{2}-4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \left(x-2\right)^{2} times \frac{6}{6}.

$$\frac{-x^{2}+4x+12}{6}-\frac{6\left(x-2\right)^{2}}{6}-4$$

Since \frac{-x^{2}+4x+12}{6} and \frac{6\left(x-2\right)^{2}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-x^{2}+4x+12-6\left(x-2\right)^{2}}{6}-4$$

Do the multiplications in -x^{2}+4x+12-6\left(x-2\right)^{2}.

$$\frac{-x^{2}+4x+12-6x^{2}+24x-24}{6}-4$$

Combine like terms in -x^{2}+4x+12-6x^{2}+24x-24.

$$\frac{-7x^{2}+28x-12}{6}-4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{6}{6}.

$$\frac{-7x^{2}+28x-12}{6}-\frac{4\times 6}{6}$$

Since \frac{-7x^{2}+28x-12}{6} and \frac{4\times 6}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-7x^{2}+28x-12-4\times 6}{6}$$

Do the multiplications in -7x^{2}+28x-12-4\times 6.

$$\frac{-7x^{2}+28x-12-24}{6}$$

Combine like terms in -7x^{2}+28x-12-24.

$$\frac{-7x^{2}+28x-36}{6}$$

Expand

$$-\frac{7x^{2}}{6}+\frac{14x}{3}-6$$

Short Solution Steps

Express x\times \frac{x+2}{3} as a single fraction.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{x\left(x+2\right)}{3} times \frac{2}{2}. Multiply \frac{x^{2}-4}{2} times \frac{3}{3}.

$$\frac{2x\left(x+2\right)}{6}-\frac{3\left(x^{2}-4\right)}{6}-\left(x-2\right)^{2}-4$$

Since \frac{2x\left(x+2\right)}{6} and \frac{3\left(x^{2}-4\right)}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x\left(x+2\right)-3\left(x^{2}-4\right)}{6}-\left(x-2\right)^{2}-4$$

Do the multiplications in 2x\left(x+2\right)-3\left(x^{2}-4\right).

$$\frac{2x^{2}+4x-3x^{2}+12}{6}-\left(x-2\right)^{2}-4$$

Combine like terms in 2x^{2}+4x-3x^{2}+12.

$$\frac{-x^{2}+4x+12}{6}-\left(x-2\right)^{2}-4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \left(x-2\right)^{2} times \frac{6}{6}.

$$\frac{-x^{2}+4x+12}{6}-\frac{6\left(x-2\right)^{2}}{6}-4$$

Since \frac{-x^{2}+4x+12}{6} and \frac{6\left(x-2\right)^{2}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-x^{2}+4x+12-6\left(x-2\right)^{2}}{6}-4$$

Do the multiplications in -x^{2}+4x+12-6\left(x-2\right)^{2}.

$$\frac{-x^{2}+4x+12-6x^{2}+24x-24}{6}-4$$

Combine like terms in -x^{2}+4x+12-6x^{2}+24x-24.

$$\frac{-7x^{2}+28x-12}{6}-4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{6}{6}.

$$\frac{-7x^{2}+28x-12}{6}-\frac{4\times 6}{6}$$

Since \frac{-7x^{2}+28x-12}{6} and \frac{4\times 6}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-7x^{2}+28x-12-4\times 6}{6}$$

Do the multiplications in -7x^{2}+28x-12-4\times 6.

$$\frac{-7x^{2}+28x-12-24}{6}$$

Combine like terms in -7x^{2}+28x-12-24.

$$\frac{-7x^{2}+28x-36}{6}$$