Solve for x (x-2)^2-(3-x)^2=1

${(x - 2)}^{2} - {(3 - x)}^{2} = 1$

Step 1

Subtract $1$ from both sides of the equation.

${(x - 2)}^{2} - {(3 - x)}^{2} - 1 = 0$

Step 2

Simplify ${(x - 2)}^{2} - {(3 - x)}^{2} - 1$ .

Simplify each term.

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(x - 2) (x - 2) - {(3 - x)}^{2} - 1 = 0$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$x (x - 2) - 2 (x - 2) - {(3 - x)}^{2} - 1 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) - {(3 - x)}^{2} - 1 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - {(3 - x)}^{2} - 1 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - {(3 - x)}^{2} - 1 = 0$

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - {(3 - x)}^{2} - 1 = 0$

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4 - {(3 - x)}^{2} - 1 = 0$

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 4 x + 4 - {(3 - x)}^{2} - 1 = 0$

Rewrite ${(3 - x)}^{2}$ as $(3 - x) (3 - x)$ .

$x^{2} - 4 x + 4 - ((3 - x) (3 - x)) - 1 = 0$

Expand $(3 - x) (3 - x)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 4 x + 4 - (3 (3 - x) - x (3 - x)) - 1 = 0$

Apply the distributive property.

$x^{2} - 4 x + 4 - (3 \cdot 3 + 3 (- x) - x (3 - x)) - 1 = 0$

Apply the distributive property.

$x^{2} - 4 x + 4 - (3 \cdot 3 + 3 (- x) - x \cdot 3 - x (- x)) - 1 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$x^{2} - 4 x + 4 - (9 + 3 (- x) - x \cdot 3 - x (- x)) - 1 = 0$

Multiply $- 1$ by $3$ .

$x^{2} - 4 x + 4 - (9 - 3 x - x \cdot 3 - x (- x)) - 1 = 0$

Multiply $3$ by $- 1$ .

$x^{2} - 4 x + 4 - (9 - 3 x - 3 x - x (- x)) - 1 = 0$

Rewrite using the commutative property of multiplication.

$x^{2} - 4 x + 4 - (9 - 3 x - 3 x - 1 \cdot - 1 x \cdot x) - 1 = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} - 4 x + 4 - (9 - 3 x - 3 x - 1 \cdot - 1 (x \cdot x)) - 1 = 0$

Multiply $x$ by $x$ .

$x^{2} - 4 x + 4 - (9 - 3 x - 3 x - 1 \cdot - 1 x^{2}) - 1 = 0$

Multiply $- 1$ by $- 1$ .

$x^{2} - 4 x + 4 - (9 - 3 x - 3 x + 1 x^{2}) - 1 = 0$

Multiply $x^{2}$ by $1$ .

$x^{2} - 4 x + 4 - (9 - 3 x - 3 x + x^{2}) - 1 = 0$

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 4 x + 4 - (9 - 6 x + x^{2}) - 1 = 0$

Apply the distributive property.

$x^{2} - 4 x + 4 - 1 \cdot 9 - (- 6 x) - x^{2} - 1 = 0$

Simplify.

Multiply $- 1$ by $9$ .

$x^{2} - 4 x + 4 - 9 - (- 6 x) - x^{2} - 1 = 0$

Multiply $- 6$ by $- 1$ .

$x^{2} - 4 x + 4 - 9 + 6 x - x^{2} - 1 = 0$

Combine the opposite terms in $x^{2} - 4 x + 4 - 9 + 6 x - x^{2} - 1$ .

Subtract $x^{2}$ from $x^{2}$ .

$- 4 x + 4 - 9 + 6 x + 0 - 1 = 0$

Add $- 4 x + 4 - 9 + 6 x$ and $0$ .

$- 4 x + 4 - 9 + 6 x - 1 = 0$

Add $- 4 x$ and $6 x$ .

$2 x + 4 - 9 - 1 = 0$

Subtract $9$ from $4$ .

$2 x - 5 - 1 = 0$

Subtract $1$ from $- 5$ .

$2 x - 6 = 0$

Step 3

Factor $2$ out of $2 x - 6$ .

Factor $2$ out of $2 x$ .

$2 (x) - 6 = 0$

Factor $2$ out of $- 6$ .

$2 x + 2 \cdot - 3 = 0$

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$2 (x - 3) = 0$

Step 4

Divide each term in $2 (x - 3) = 0$ by $2$ and simplify.

Divide each term in $2 (x - 3) = 0$ by $2$ .

$\frac{2 (x - 3)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (x - 3)}{2} = \frac{0}{2}$

Divide $x - 3$ by $1$ .

$x - 3 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x - 3 = 0$

Step 5

Add $3$ to both sides of the equation.

$x = 3$

Solve for x (x-2)^2-(3-x)^2=1

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