Solve for x (13x-1)^2-2(13x-1)-3=0

${(13 x - 1)}^{2} - 2 (13 x - 1) - 3 = 0$

Factor the left side of the equation.

Let $u = 13 x - 1$ . Substitute $u$ for all occurrences of $13 x - 1$ .

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

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

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$(u - 3) (u + 1) = 0$

Replace all occurrences of $u$ with $13 x - 1$ .

$(13 x - 1 - 3) (13 x - 1 + 1) = 0$

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

$13 x - 1 - 3 = 0$

$13 x - 1 + 1 = 0$

Set $13 x - 1 - 3$ equal to $0$ and solve for $x$ .

Set $13 x - 1 - 3$ equal to $0$ .

$13 x - 1 - 3 = 0$

Solve $13 x - 1 - 3 = 0$ for $x$ .

Subtract $3$ from $- 1$ .

$13 x - 4 = 0$

Add $4$ to both sides of the equation.

$13 x = 4$

Divide each term in $13 x = 4$ by $13$ and simplify.

Divide each term in $13 x = 4$ by $13$ .

$\frac{13 x}{13} = \frac{4}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 x}{13} = \frac{4}{13}$

Divide $x$ by $1$ .

$x = \frac{4}{13}$

Set $13 x - 1 + 1$ equal to $0$ and solve for $x$ .

Set $13 x - 1 + 1$ equal to $0$ .

$13 x - 1 + 1 = 0$

Solve $13 x - 1 + 1 = 0$ for $x$ .

Combine the opposite terms in $13 x - 1 + 1$ .

Add $- 1$ and $1$ .

$13 x + 0 = 0$

Add $13 x$ and $0$ .

$13 x = 0$

Divide each term in $13 x = 0$ by $13$ and simplify.

Divide each term in $13 x = 0$ by $13$ .

$\frac{13 x}{13} = \frac{0}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 x}{13} = \frac{0}{13}$

Divide $x$ by $1$ .

$x = \frac{0}{13}$

Simplify the right side.

Divide $0$ by $13$ .

$x = 0$

The final solution is all the values that make $(13 x - 1 - 3) (13 x - 1 + 1) = 0$ true.

$x = \frac{4}{13}, 0$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{13}, 0$

Decimal Form:

$x = 0.\overline{307692}, 0$

Solve for x (13x-1)^2-2(13x-1)-3=0

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