Pre-Algebra Examples

$(3 x - 10) (x - 4) = 3 (x - 2) + 18$

Step 1

Simplify $(3 x - 10) (x - 4)$ .

Tap for more steps...

Step 1.1

Rewrite.

$0 + 0 + (3 x - 10) (x - 4) = 3 (x - 2) + 18$

Step 1.2

Simplify by adding zeros.

$(3 x - 10) (x - 4) = 3 (x - 2) + 18$

Step 1.3

Expand $(3 x - 10) (x - 4)$ using the FOIL Method.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

$3 x (x - 4) - 10 (x - 4) = 3 (x - 2) + 18$

Step 1.3.2

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 4 - 10 (x - 4) = 3 (x - 2) + 18$

Step 1.3.3

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 4 - 10 x - 10 \cdot - 4 = 3 (x - 2) + 18$

Step 1.4

Simplify and combine like terms.

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

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

Tap for more steps...

Step 1.4.1.1.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 4 - 10 x - 10 \cdot - 4 = 3 (x - 2) + 18$

Step 1.4.1.1.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 4 - 10 x - 10 \cdot - 4 = 3 (x - 2) + 18$

Step 1.4.1.2

Multiply $- 4$ by $3$ .

$3 x^{2} - 12 x - 10 x - 10 \cdot - 4 = 3 (x - 2) + 18$

Step 1.4.1.3

Multiply $- 10$ by $- 4$ .

$3 x^{2} - 12 x - 10 x + 40 = 3 (x - 2) + 18$

Step 1.4.2

Subtract $10 x$ from $- 12 x$ .

$3 x^{2} - 22 x + 40 = 3 (x - 2) + 18$

Step 2

Simplify $3 (x - 2) + 18$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Apply the distributive property.

$3 x^{2} - 22 x + 40 = 3 x + 3 \cdot - 2 + 18$

Step 2.1.2

Multiply $3$ by $- 2$ .

$3 x^{2} - 22 x + 40 = 3 x - 6 + 18$

Step 2.2

Add $- 6$ and $18$ .

$3 x^{2} - 22 x + 40 = 3 x + 12$

Step 3

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

Tap for more steps...

Step 3.1

Subtract $3 x$ from both sides of the equation.

$3 x^{2} - 22 x + 40 - 3 x = 12$

Step 3.2

Subtract $3 x$ from $- 22 x$ .

$3 x^{2} - 25 x + 40 = 12$

Step 4

Subtract $12$ from both sides of the equation.

$3 x^{2} - 25 x + 40 - 12 = 0$

Step 5

Subtract $12$ from $40$ .

$3 x^{2} - 25 x + 28 = 0$

Step 6

Factor by grouping.

Tap for more steps...

Step 6.1

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 $a \cdot c = 3 \cdot 28 = 84$ and whose sum is $b = - 25$ .

Tap for more steps...

Step 6.1.1

Factor $- 25$ out of $- 25 x$ .

$3 x^{2} - 25 x + 28 = 0$

Step 6.1.2

Rewrite $- 25$ as $- 4$ plus $- 21$

$3 x^{2} + (- 4 - 21) x + 28 = 0$

Step 6.1.3

Apply the distributive property.

$3 x^{2} - 4 x - 21 x + 28 = 0$

Step 6.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 6.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 4 x) - 21 x + 28 = 0$

Step 6.2.2

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

$x (3 x - 4) - 7 (3 x - 4) = 0$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 4$ .

$(3 x - 4) (x - 7) = 0$

Step 7

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

$3 x - 4 = 0$

$x - 7 = 0$

Step 8

Set $3 x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 8.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 8.2

Solve $3 x - 4 = 0$ for $x$ .

Tap for more steps...

Step 8.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 8.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

Tap for more steps...

Step 8.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 8.2.2.2

Simplify the left side.

Tap for more steps...

Step 8.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 9

Set $x - 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 9.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 9.2

Add $7$ to both sides of the equation.

$x = 7$

Step 10

The final solution is all the values that make $(3 x - 4) (x - 7) = 0$ true.

$x = \frac{4}{3}, 7$