Finite Math Examples

$5 x (10 - 3 x) = 15$

Step 1

Divide each term in $5 x (10 - 3 x) = 15$ by $5$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $5 x (10 - 3 x) = 15$ by $5$ .

$\frac{5 x (10 - 3 x)}{5} = \frac{15}{5}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Simplify terms.

Tap for more steps...

Step 1.2.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.2.1.1.1

Cancel the common factor.

$\frac{5 x (10 - 3 x)}{5} = \frac{15}{5}$

Step 1.2.1.1.2

Divide $x (10 - 3 x)$ by $1$ .

$x (10 - 3 x) = \frac{15}{5}$

Step 1.2.1.2

Apply the distributive property.

$x \cdot 10 + x (- 3 x) = \frac{15}{5}$

Step 1.2.1.3

Reorder.

Tap for more steps...

Step 1.2.1.3.1

Move $10$ to the left of $x$ .

$10 \cdot x + x (- 3 x) = \frac{15}{5}$

Step 1.2.1.3.2

Rewrite using the commutative property of multiplication.

$10 \cdot x - 3 x \cdot x = \frac{15}{5}$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

Move $x$ .

$10 x - 3 (x \cdot x) = \frac{15}{5}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$10 x - 3 x^{2} = \frac{15}{5}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Divide $15$ by $5$ .

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

Step 2

Subtract $3$ from both sides of the equation.

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

Step 3

Factor the left side of the equation.

Tap for more steps...

Step 3.1

Factor $- 1$ out of $10 x - 3 x^{2} - 3$ .

Tap for more steps...

Step 3.1.1

Reorder $10 x$ and $- 3 x^{2}$ .

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

Step 3.1.2

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 3.1.3

Factor $- 1$ out of $10 x$ .

$- (3 x^{2}) - (- 10 x) - 3 = 0$

Step 3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$- (3 x^{2}) - (- 10 x) - 1 \cdot 3 = 0$

Step 3.1.5

Factor $- 1$ out of $- (3 x^{2}) - (- 10 x)$ .

$- (3 x^{2} - 10 x) - 1 \cdot 3 = 0$

Step 3.1.6

Factor $- 1$ out of $- (3 x^{2} - 10 x) - 1 (3)$ .

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

Step 3.2

Factor.

Tap for more steps...

Step 3.2.1

Factor by grouping.

Tap for more steps...

Step 3.2.1.1

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

Tap for more steps...

Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

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

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

Step 3.2.1.1.3

Apply the distributive property.

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

Step 3.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1.2.1

Group the first two terms and the last two terms.

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

Step 3.2.1.2.2

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

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

Step 3.2.1.3

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

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

Step 3.2.2

Remove unnecessary parentheses.

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

Step 4

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

$3 x - 1 = 0$

$x - 3 = 0$

Step 5

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

Tap for more steps...

Step 5.1

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

$3 x - 1 = 0$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 5.2.2

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

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

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

Tap for more steps...

Step 6.1

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

$x - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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