Solve (x+y+z)(x-y+z)-(x+y-z)(x-y-z) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$(x+y+z)(x-y+z)-(x+y-z)(x-y-z)$$

Evaluate

$4xz$

Solution Steps

Apply the distributive property by multiplying each term of $x+y+z$ by each term of $x-y+z$.

$$x^{2}-xy+xz+yx-y^{2}+yz+zx-zy+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Combine $-xy$ and $yx$ to get $0$.

$$x^{2}+xz-y^{2}+yz+zx-zy+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Combine $xz$ and $zx$ to get $2xz$.

$$x^{2}+2xz-y^{2}+yz-zy+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Combine $yz$ and $-zy$ to get $0$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Apply the distributive property by multiplying each term of $x+y-z$ by each term of $x-y-z$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-xy-xz+yx-y^{2}-yz-zx+zy+z^{2}\right)$$

Combine $-xy$ and $yx$ to get $0$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-xz-y^{2}-yz-zx+zy+z^{2}\right)$$

Combine $-xz$ and $-zx$ to get $-2xz$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-2xz-y^{2}-yz+zy+z^{2}\right)$$

Combine $-yz$ and $zy$ to get $0$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-2xz-y^{2}+z^{2}\right)$$

To find the opposite of $x^{2}-2xz-y^{2}+z^{2}$, find the opposite of each term.

$$x^{2}+2xz-y^{2}+z^{2}-x^{2}-\left(-2xz\right)-\left(-y^{2}\right)-z^{2}$$

The opposite of $-2xz$ is $2xz$.

$$x^{2}+2xz-y^{2}+z^{2}-x^{2}+2xz-\left(-y^{2}\right)-z^{2}$$

The opposite of $-y^{2}$ is $y^{2}$.

$$x^{2}+2xz-y^{2}+z^{2}-x^{2}+2xz+y^{2}-z^{2}$$

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

$$2xz-y^{2}+z^{2}+2xz+y^{2}-z^{2}$$

Combine $2xz$ and $2xz$ to get $4xz$.

$$4xz-y^{2}+z^{2}+y^{2}-z^{2}$$

Combine $-y^{2}$ and $y^{2}$ to get $0$.

$$4xz+z^{2}-z^{2}$$

Combine $z^{2}$ and $-z^{2}$ to get $0$.

$$4xz$$

Expand

$4xz$

Solution Steps

Apply the distributive property by multiplying each term of $x+y+z$ by each term of $x-y+z$.

$$x^{2}-xy+xz+yx-y^{2}+yz+zx-zy+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Combine $-xy$ and $yx$ to get $0$.

$$x^{2}+xz-y^{2}+yz+zx-zy+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Combine $xz$ and $zx$ to get $2xz$.

$$x^{2}+2xz-y^{2}+yz-zy+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Combine $yz$ and $-zy$ to get $0$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x+y-z\right)\left(x-y-z\right)$$

Apply the distributive property by multiplying each term of $x+y-z$ by each term of $x-y-z$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-xy-xz+yx-y^{2}-yz-zx+zy+z^{2}\right)$$

Combine $-xy$ and $yx$ to get $0$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-xz-y^{2}-yz-zx+zy+z^{2}\right)$$

Combine $-xz$ and $-zx$ to get $-2xz$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-2xz-y^{2}-yz+zy+z^{2}\right)$$

Combine $-yz$ and $zy$ to get $0$.

$$x^{2}+2xz-y^{2}+z^{2}-\left(x^{2}-2xz-y^{2}+z^{2}\right)$$

To find the opposite of $x^{2}-2xz-y^{2}+z^{2}$, find the opposite of each term.

$$x^{2}+2xz-y^{2}+z^{2}-x^{2}-\left(-2xz\right)-\left(-y^{2}\right)-z^{2}$$

The opposite of $-2xz$ is $2xz$.

$$x^{2}+2xz-y^{2}+z^{2}-x^{2}+2xz-\left(-y^{2}\right)-z^{2}$$

The opposite of $-y^{2}$ is $y^{2}$.

$$x^{2}+2xz-y^{2}+z^{2}-x^{2}+2xz+y^{2}-z^{2}$$

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

$$2xz-y^{2}+z^{2}+2xz+y^{2}-z^{2}$$

Combine $2xz$ and $2xz$ to get $4xz$.

$$4xz-y^{2}+z^{2}+y^{2}-z^{2}$$

Combine $-y^{2}$ and $y^{2}$ to get $0$.

$$4xz+z^{2}-z^{2}$$

Combine $z^{2}$ and $-z^{2}$ to get $0$.

$$4xz$$