Precalculus Examples

$4 n^{2} = 25$

Step 1

Subtract $25$ from both sides of the equation.

$4 n^{2} - 25 = 0$

Step 2

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

${(2 n)}^{2} - 25 = 0$

Step 3

Rewrite $25$ as $5^{2}$ .

${(2 n)}^{2} - 5^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 n$ and $b = 5$ .

$(2 n + 5) (2 n - 5) = 0$

Step 5

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

$2 n + 5 = 0$

$2 n - 5 = 0$

Step 6

Set $2 n + 5$ equal to $0$ and solve for $n$ .

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Step 6.1

Set $2 n + 5$ equal to $0$ .

$2 n + 5 = 0$

Step 6.2

Solve $2 n + 5 = 0$ for $n$ .

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Step 6.2.1

Subtract $5$ from both sides of the equation.

$2 n = - 5$

Step 6.2.2

Divide each term in $2 n = - 5$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 n = - 5$ by $2$ .

$\frac{2 n}{2} = \frac{- 5}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 n}{2} = \frac{- 5}{2}$

Step 6.2.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 5}{2}$

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.

$n = - \frac{5}{2}$

Step 7

Set $2 n - 5$ equal to $0$ and solve for $n$ .

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Step 7.1

Set $2 n - 5$ equal to $0$ .

$2 n - 5 = 0$

Step 7.2

Solve $2 n - 5 = 0$ for $n$ .

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Step 7.2.1

Add $5$ to both sides of the equation.

$2 n = 5$

Step 7.2.2

Divide each term in $2 n = 5$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 n = 5$ by $2$ .

$\frac{2 n}{2} = \frac{5}{2}$

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.

$\frac{2 n}{2} = \frac{5}{2}$

Step 7.2.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{5}{2}$

Step 8

The final solution is all the values that make $(2 n + 5) (2 n - 5) = 0$ true.

$n = - \frac{5}{2}, \frac{5}{2}$