Solve for x (3x-4)(x-2)-5=0

$(3 x - 4) (x - 2) - 5 = 0$

Step 1

Simplify $(3 x - 4) (x - 2) - 5$ .

Simplify each term.

Expand $(3 x - 4) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$3 x (x - 2) - 4 (x - 2) - 5 = 0$

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 2 - 4 (x - 2) - 5 = 0$

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 2 - 4 x - 4 \cdot - 2 - 5 = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 2 - 4 x - 4 \cdot - 2 - 5 = 0$

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 2 - 4 x - 4 \cdot - 2 - 5 = 0$

Multiply $- 2$ by $3$ .

$3 x^{2} - 6 x - 4 x - 4 \cdot - 2 - 5 = 0$

Multiply $- 4$ by $- 2$ .

$3 x^{2} - 6 x - 4 x + 8 - 5 = 0$

Subtract $4 x$ from $- 6 x$ .

$3 x^{2} - 10 x + 8 - 5 = 0$

Subtract $5$ from $8$ .

$3 x^{2} - 10 x + 3 = 0$

Step 2

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 $a \cdot c = 3 \cdot 3 = 9$ and whose sum is $b = - 10$ .

Factor $- 10$ out of $- 10 x$ .

$3 x^{2} - 10 x + 3 = 0$

Rewrite $- 10$ as $- 1$ plus $- 9$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$x (3 x - 1) - 3 (3 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x - 3) = 0$

Step 3

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

$3 x - 1 = 0$

$x - 3 = 0$

Step 4

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 5

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 6

The final solution is all the values that make $(3 x - 1) (x - 3) = 0$ true.

$x = \frac{1}{3}, 3$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, 3$

Decimal Form:

$x = 0.\overline{3}, 3$

Solve for x (3x-4)(x-2)-5=0

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