Algebra Examples

${(17 x - 1)}^{2} - 2 (17 x - 1) - 3 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = 17 x - 1$ . Substitute $u$ for all occurrences of $17 x - 1$ .

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

Step 1.2

Factor $u^{2} - 2 u - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

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

Step 1.3

Replace all occurrences of $u$ with $17 x - 1$ .

$(17 x - 1 - 3) (17 x - 1 + 1) = 0$

Step 2

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

$17 x - 1 - 3 = 0$

$17 x - 1 + 1 = 0$

Step 3

Set $17 x - 1 - 3$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $17 x - 1 - 3$ equal to $0$ .

$17 x - 1 - 3 = 0$

Step 3.2

Solve $17 x - 1 - 3 = 0$ for $x$ .

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Step 3.2.1

Subtract $3$ from $- 1$ .

$17 x - 4 = 0$

Step 3.2.2

Add $4$ to both sides of the equation.

$17 x = 4$

Step 3.2.3

Divide each term in $17 x = 4$ by $17$ and simplify.

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Step 3.2.3.1

Divide each term in $17 x = 4$ by $17$ .

$\frac{17 x}{17} = \frac{4}{17}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $17$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{17 x}{17} = \frac{4}{17}$

Step 3.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{17}$

Step 4

Set $17 x - 1 + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $17 x - 1 + 1$ equal to $0$ .

$17 x - 1 + 1 = 0$

Step 4.2

Solve $17 x - 1 + 1 = 0$ for $x$ .

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Step 4.2.1

Combine the opposite terms in $17 x - 1 + 1$ .

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Step 4.2.1.1

Add $- 1$ and $1$ .

$17 x + 0 = 0$

Step 4.2.1.2

Add $17 x$ and $0$ .

$17 x = 0$

Step 4.2.2

Divide each term in $17 x = 0$ by $17$ and simplify.

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Step 4.2.2.1

Divide each term in $17 x = 0$ by $17$ .

$\frac{17 x}{17} = \frac{0}{17}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $17$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{17 x}{17} = \frac{0}{17}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{17}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $0$ by $17$ .

$x = 0$

Step 5

The final solution is all the values that make $(17 x - 1 - 3) (17 x - 1 + 1) = 0$ true.

$x = \frac{4}{17}, 0$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{17}, 0$

Decimal Form:

$x = 0.23529411 \dots, 0$