Finite Math Examples

$16 n^{2} = 49$

Step 1

Subtract $49$ from both sides of the equation.

$16 n^{2} - 49 = 0$

Step 2

Rewrite $16 n^{2}$ as ${(4 n)}^{2}$ .

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

Step 3

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

${(4 n)}^{2} - 7^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 n$ and $b = 7$ .

$(4 n + 7) (4 n - 7) = 0$

Step 5

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

$4 n + 7 = 0$

$4 n - 7 = 0$

Step 6

Set $4 n + 7$ equal to $0$ and solve for $n$ .

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Step 6.1

Set $4 n + 7$ equal to $0$ .

$4 n + 7 = 0$

Step 6.2

Solve $4 n + 7 = 0$ for $n$ .

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Step 6.2.1

Subtract $7$ from both sides of the equation.

$4 n = - 7$

Step 6.2.2

Divide each term in $4 n = - 7$ by $4$ and simplify.

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Step 6.2.2.1

Divide each term in $4 n = - 7$ by $4$ .

$\frac{4 n}{4} = \frac{- 7}{4}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{4 n}{4} = \frac{- 7}{4}$

Step 6.2.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 7}{4}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$n = - \frac{7}{4}$

Step 7

Set $4 n - 7$ equal to $0$ and solve for $n$ .

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Step 7.1

Set $4 n - 7$ equal to $0$ .

$4 n - 7 = 0$

Step 7.2

Solve $4 n - 7 = 0$ for $n$ .

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Step 7.2.1

Add $7$ to both sides of the equation.

$4 n = 7$

Step 7.2.2

Divide each term in $4 n = 7$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 n = 7$ by $4$ .

$\frac{4 n}{4} = \frac{7}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 n}{4} = \frac{7}{4}$

Step 7.2.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{7}{4}$

Step 8

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

$n = - \frac{7}{4}, \frac{7}{4}$