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

$$(x-2)(x+2)-(x-1)(x-3)$$

Evaluate

$$4x-7$$

Solution Steps

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}.

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

Calculate 2 to the power of 2 and get 4.

$$x^{2}-4-\left(x-1\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x-1 by each term of x-3.

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

Combine -3x and -x to get -4x.

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

To find the opposite of x^{2}-4x+3, find the opposite of each term.

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

The opposite of -4x is 4x.

$$x^{2}-4-x^{2}+4x-3$$

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

$$-4+4x-3$$

Subtract 3 from -4 to get -7.

$$-7+4x$$

Expand

$$4x-7$$

Solution Steps

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}.

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

Calculate 2 to the power of 2 and get 4.

$$x^{2}-4-\left(x-1\right)\left(x-3\right)$$

Apply the distributive property by multiplying each term of x-1 by each term of x-3.

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

Combine -3x and -x to get -4x.

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

To find the opposite of x^{2}-4x+3, find the opposite of each term.

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

The opposite of -4x is 4x.

$$x^{2}-4-x^{2}+4x-3$$

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

$$-4+4x-3$$

Subtract 3 from -4 to get -7.

$$-7+4x$$