Finite Math Examples

$2 x (5 x - 7) = 2 x^{2} - 5$

Step 1

Simplify $2 x (5 x - 7)$ .

Tap for more steps...

Step 1.1

Rewrite.

$0 + 0 + 2 x (5 x - 7) = 2 x^{2} - 5$

Step 1.2

Simplify by multiplying through.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$2 x (5 x) + 2 x \cdot - 7 = 2 x^{2} - 5$

Step 1.2.2

Simplify the expression.

Tap for more steps...

Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 5 x \cdot x + 2 x \cdot - 7 = 2 x^{2} - 5$

Step 1.2.2.2

Multiply $- 7$ by $2$ .

$2 \cdot 5 x \cdot x - 14 x = 2 x^{2} - 5$

Step 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

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

Tap for more steps...

Step 1.3.1.1

Move $x$ .

$2 \cdot 5 (x \cdot x) - 14 x = 2 x^{2} - 5$

Step 1.3.1.2

Multiply $x$ by $x$ .

$2 \cdot 5 x^{2} - 14 x = 2 x^{2} - 5$

Step 1.3.2

Multiply $2$ by $5$ .

$10 x^{2} - 14 x = 2 x^{2} - 5$

Step 2

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

Tap for more steps...

Step 2.1

Subtract $2 x^{2}$ from both sides of the equation.

$10 x^{2} - 14 x - 2 x^{2} = - 5$

Step 2.2

Subtract $2 x^{2}$ from $10 x^{2}$ .

$8 x^{2} - 14 x = - 5$

Step 3

Add $5$ to both sides of the equation.

$8 x^{2} - 14 x + 5 = 0$

Step 4

Factor by grouping.

Tap for more steps...

Step 4.1

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 $a \cdot c = 8 \cdot 5 = 40$ and whose sum is $b = - 14$ .

Tap for more steps...

Step 4.1.1

Factor $- 14$ out of $- 14 x$ .

$8 x^{2} - 14 x + 5 = 0$

Step 4.1.2

Rewrite $- 14$ as $- 4$ plus $- 10$

$8 x^{2} + (- 4 - 10) x + 5 = 0$

Step 4.1.3

Apply the distributive property.

$8 x^{2} - 4 x - 10 x + 5 = 0$

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

$(8 x^{2} - 4 x) - 10 x + 5 = 0$

Step 4.2.2

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

$4 x (2 x - 1) - 5 (2 x - 1) = 0$

Step 4.3

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

$(2 x - 1) (4 x - 5) = 0$

Step 5

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

$2 x - 1 = 0$

$4 x - 5 = 0$

Step 6

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

Tap for more steps...

Step 6.1

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

$2 x - 1 = 0$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 6.2.2

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

Tap for more steps...

Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 7

Set $4 x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Step 7.2

Solve $4 x - 5 = 0$ for $x$ .

Tap for more steps...

Step 7.2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 7.2.2

Divide each term in $4 x = 5$ by $4$ and simplify.

Tap for more steps...

Step 7.2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 7.2.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 8

The final solution is all the values that make $(2 x - 1) (4 x - 5) = 0$ true.

$x = \frac{1}{2}, \frac{5}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, \frac{5}{4}$

Decimal Form:

$x = 0.5, 1.25$

Mixed Number Form:

$x = \frac{1}{2}, 1 \frac{1}{4}$