Basic Math Examples

${(4 n + 1)}^{2} + 3 (4 n + 1) - 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = 4 n + 1$ . Substitute $u$ for all occurrences of $4 n + 1$ .

$u^{2} + 3 u - 4 = 0$

Step 1.2

Factor $u^{2} + 3 u - 4$ using the AC method.

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Step 1.2.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.2.2

Write the factored form using these integers.

$(u - 1) (u + 4) = 0$

Step 1.3

Replace all occurrences of $u$ with $4 n + 1$ .

$(4 n + 1 - 1) (4 n + 1 + 4) = 0$

Step 2

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

$4 n + 1 - 1 = 0$

$4 n + 1 + 4 = 0$

Step 3

Set $4 n + 1 - 1$ equal to $0$ and solve for $n$ .

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Step 3.1

Set $4 n + 1 - 1$ equal to $0$ .

$4 n + 1 - 1 = 0$

Step 3.2

Solve $4 n + 1 - 1 = 0$ for $n$ .

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Step 3.2.1

Combine the opposite terms in $4 n + 1 - 1$ .

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Step 3.2.1.1

Subtract $1$ from $1$ .

$4 n + 0 = 0$

Step 3.2.1.2

Add $4 n$ and $0$ .

$4 n = 0$

Step 3.2.2

Divide each term in $4 n = 0$ by $4$ and simplify.

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Step 3.2.2.1

Divide each term in $4 n = 0$ by $4$ .

$\frac{4 n}{4} = \frac{0}{4}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{4 n}{4} = \frac{0}{4}$

Step 3.2.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{0}{4}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $0$ by $4$ .

$n = 0$

Step 4

Set $4 n + 1 + 4$ equal to $0$ and solve for $n$ .

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Step 4.1

Set $4 n + 1 + 4$ equal to $0$ .

$4 n + 1 + 4 = 0$

Step 4.2

Solve $4 n + 1 + 4 = 0$ for $n$ .

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Step 4.2.1

Add $1$ and $4$ .

$4 n + 5 = 0$

Step 4.2.2

Subtract $5$ from both sides of the equation.

$4 n = - 5$

Step 4.2.3

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

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Step 4.2.3.1

Divide each term in $4 n = - 5$ by $4$ .

$\frac{4 n}{4} = \frac{- 5}{4}$

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of $4$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

$\frac{4 n}{4} = \frac{- 5}{4}$

Step 4.2.3.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 5}{4}$

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Move the negative in front of the fraction.

$n = - \frac{5}{4}$

Step 5

The final solution is all the values that make $(4 n + 1 - 1) (4 n + 1 + 4) = 0$ true.

$n = 0, - \frac{5}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$n = 0, - \frac{5}{4}$

Decimal Form:

$n = 0, - 1.25$

Mixed Number Form:

$n = 0, - 1 \frac{1}{4}$