Solve Using the Quadratic Formula (x-2)(x-3)=2

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

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Simplify $(x - 2) (x - 3)$ .

Expand $(x - 2) (x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 3$ .

$x^{2} - 3 x - 2 x + 6 = 2$

Subtract $2 x$ from $- 3 x$ .

$x^{2} - 5 x + 6 = 2$

Subtract $2$ from both sides of the equation.

$x^{2} - 5 x + 6 - 2 = 0$

Subtract $2$ from $6$ .

$x^{2} - 5 x + 4 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 5$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$x = \frac{5 \pm \sqrt{25 - 16}}{2 \cdot 1}$

Subtract $16$ from $25$ .

$x = \frac{5 \pm \sqrt{9}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm \sqrt{3^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{5 \pm 3}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 4, 1$

Solve Using the Quadratic Formula (x-2)(x-3)=2

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