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

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

Step 1

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) - 2 - x = 1$

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 2 - x = 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 - 2 - x = 1$

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

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

Subtract $x$ from $- 6 x$ .

$x^{2} - 7 x + 9 - 2 = 1$

Subtract $2$ from $9$ .

$x^{2} - 7 x + 7 = 1$

Step 2

Subtract $1$ from both sides of the equation.

$x^{2} - 7 x + 7 - 1 = 0$

Step 3

Subtract $1$ from $7$ .

$x^{2} - 7 x + 6 = 0$

Step 4

Factor $x^{2} - 7 x + 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 7$ .

$- 6, - 1$

Write the factored form using these integers.

$(x - 6) (x - 1) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x - 1 = 0$

Step 6

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 7

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 8

The final solution is all the values that make $(x - 6) (x - 1) = 0$ true.

$x = 6, 1$

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

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