Solve by Factoring 2x^2-3x=2x-2

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

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

Add $2$ to both sides of the equation.

$2 x^{2} - 3 x - 2 x + 2 = 0$

Step 2

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

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

Step 3

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

$x (2 x - 1) - 2 (2 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x - 1) (x - 2) = 0$

Step 4

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

$2 x - 1 = 0$

$x - 2 = 0$

Step 5

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $(2 x - 1) (x - 2) = 0$ true.

$x = \frac{1}{2}, 2$

Solve by Factoring 2x^2-3x=2x-2

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