Finite Math Examples

$12 x^{2} = 147$

Step 1

Subtract $147$ from both sides of the equation.

$12 x^{2} - 147 = 0$

Step 2

Factor $3$ out of $12 x^{2} - 147$ .

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Step 2.1

Factor $3$ out of $12 x^{2}$ .

$3 (4 x^{2}) - 147 = 0$

Step 2.2

Factor $3$ out of $- 147$ .

$3 (4 x^{2}) + 3 (- 49) = 0$

Step 2.3

Factor $3$ out of $3 (4 x^{2}) + 3 (- 49)$ .

$3 (4 x^{2} - 49) = 0$

Step 3

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$3 ({(2 x)}^{2} - 49) = 0$

Step 4

Rewrite $49$ as $7^{2}$ .

$3 ({(2 x)}^{2} - 7^{2}) = 0$

Step 5

Factor.

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Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 7$ .

$3 ((2 x + 7) (2 x - 7)) = 0$

Step 5.2

Remove unnecessary parentheses.

$3 (2 x + 7) (2 x - 7) = 0$

Step 6

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

$2 x + 7 = 0$

$2 x - 7 = 0$

Step 7

Set $2 x + 7$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Step 7.2

Solve $2 x + 7 = 0$ for $x$ .

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Step 7.2.1

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Step 7.2.2

Divide each term in $2 x = - 7$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 8

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 8.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 8.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 8.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 8.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 9

The final solution is all the values that make $3 (2 x + 7) (2 x - 7) = 0$ true.

$x = - \frac{7}{2}, \frac{7}{2}$