Finite Math Examples

$47 - 47 x = (7 x + 2) (x - 1)$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$(7 x + 2) (x - 1) = 47 - 47 x$

Step 2

Simplify $(7 x + 2) (x - 1)$ .

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Step 2.1

Rewrite.

$0 + 0 + (7 x + 2) (x - 1) = 47 - 47 x$

Step 2.2

Simplify by adding zeros.

$(7 x + 2) (x - 1) = 47 - 47 x$

Step 2.3

Expand $(7 x + 2) (x - 1)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$7 x (x - 1) + 2 (x - 1) = 47 - 47 x$

Step 2.3.2

Apply the distributive property.

$7 x \cdot x + 7 x \cdot - 1 + 2 (x - 1) = 47 - 47 x$

Step 2.3.3

Apply the distributive property.

$7 x \cdot x + 7 x \cdot - 1 + 2 x + 2 \cdot - 1 = 47 - 47 x$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

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

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Step 2.4.1.1.1

Move $x$ .

$7 (x \cdot x) + 7 x \cdot - 1 + 2 x + 2 \cdot - 1 = 47 - 47 x$

Step 2.4.1.1.2

Multiply $x$ by $x$ .

$7 x^{2} + 7 x \cdot - 1 + 2 x + 2 \cdot - 1 = 47 - 47 x$

Step 2.4.1.2

Multiply $- 1$ by $7$ .

$7 x^{2} - 7 x + 2 x + 2 \cdot - 1 = 47 - 47 x$

Step 2.4.1.3

Multiply $2$ by $- 1$ .

$7 x^{2} - 7 x + 2 x - 2 = 47 - 47 x$

Step 2.4.2

Add $- 7 x$ and $2 x$ .

$7 x^{2} - 5 x - 2 = 47 - 47 x$

Step 3

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

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Step 3.1

Add $47 x$ to both sides of the equation.

$7 x^{2} - 5 x - 2 + 47 x = 47$

Step 3.2

Add $- 5 x$ and $47 x$ .

$7 x^{2} + 42 x - 2 = 47$

Step 4

Subtract $47$ from both sides of the equation.

$7 x^{2} + 42 x - 2 - 47 = 0$

Step 5

Subtract $47$ from $- 2$ .

$7 x^{2} + 42 x - 49 = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $7$ out of $7 x^{2} + 42 x - 49$ .

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Step 6.1.1

Factor $7$ out of $7 x^{2}$ .

$7 (x^{2}) + 42 x - 49 = 0$

Step 6.1.2

Factor $7$ out of $42 x$ .

$7 (x^{2}) + 7 (6 x) - 49 = 0$

Step 6.1.3

Factor $7$ out of $- 49$ .

$7 x^{2} + 7 (6 x) + 7 \cdot - 7 = 0$

Step 6.1.4

Factor $7$ out of $7 x^{2} + 7 (6 x)$ .

$7 (x^{2} + 6 x) + 7 \cdot - 7 = 0$

Step 6.1.5

Factor $7$ out of $7 (x^{2} + 6 x) + 7 \cdot - 7$ .

$7 (x^{2} + 6 x - 7) = 0$

Step 6.2

Factor.

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Step 6.2.1

Factor $x^{2} + 6 x - 7$ using the AC method.

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Step 6.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 6.2.1.2

Write the factored form using these integers.

$7 ((x - 1) (x + 7)) = 0$

Step 6.2.2

Remove unnecessary parentheses.

$7 (x - 1) (x + 7) = 0$

Step 7

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

$x - 1 = 0$

$x + 7 = 0$

Step 8

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

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Step 8.1

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

$x - 1 = 0$

Step 8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9

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

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Step 9.1

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

$x + 7 = 0$

Step 9.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 10

The final solution is all the values that make $7 (x - 1) (x + 7) = 0$ true.

$x = 1, - 7$