Algebra Examples

${(x - \frac{30}{x})}^{2} - 6 (x - \frac{30}{x}) - 7 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = x - \frac{30}{x}$ . Substitute $u$ for all occurrences of $x - \frac{30}{x}$ .

$u^{2} - 6 u - 7 = 0$

Step 1.2

Factor $u^{2} - 6 u - 7$ 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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 1.2.2

Write the factored form using these integers.

$(u - 7) (u + 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $x - \frac{30}{x}$ .

$(x - \frac{30}{x} - 7) (x - \frac{30}{x} + 1) = 0$

Step 2

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

$x - \frac{30}{x} - 7 = 0$

$x - \frac{30}{x} + 1 = 0$

Step 3

Set $x - \frac{30}{x} - 7$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - \frac{30}{x} - 7$ equal to $0$ .

$x - \frac{30}{x} - 7 = 0$

Step 3.2

Solve $x - \frac{30}{x} - 7 = 0$ for $x$ .

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Step 3.2.1

Find the LCD of the terms in the equation.

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Step 3.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1, 1$

Step 3.2.1.2

The LCM of one and any expression is the expression.

$x$

Step 3.2.2

Multiply each term in $x - \frac{30}{x} - 7 = 0$ by $x$ to eliminate the fractions.

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Step 3.2.2.1

Multiply each term in $x - \frac{30}{x} - 7 = 0$ by $x$ .

$x \cdot x - \frac{30}{x} x - 7 x = 0 x$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Simplify each term.

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Step 3.2.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} - \frac{30}{x} x - 7 x = 0 x$

Step 3.2.2.2.1.2

Cancel the common factor of $x$ .

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Step 3.2.2.2.1.2.1

Move the leading negative in $- \frac{30}{x}$ into the numerator.

$x^{2} + \frac{- 30}{x} x - 7 x = 0 x$

Step 3.2.2.2.1.2.2

Cancel the common factor.

$x^{2} + \frac{- 30}{x} x - 7 x = 0 x$

Step 3.2.2.2.1.2.3

Rewrite the expression.

$x^{2} - 30 - 7 x = 0 x$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Multiply $0$ by $x$ .

$x^{2} - 30 - 7 x = 0$

Step 3.2.3

Solve the equation.

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Step 3.2.3.1

Factor $x^{2} - 30 - 7 x$ using the AC method.

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Step 3.2.3.1.1

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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Step 3.2.3.1.2

Write the factored form using these integers.

$(x - 10) (x + 3) = 0$

Step 3.2.3.2

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

$x - 10 = 0$

$x + 3 = 0$

Step 3.2.3.3

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 3.2.3.3.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 3.2.3.3.2

Add $10$ to both sides of the equation.

$x = 10$

Step 3.2.3.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.2.3.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.2.3.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.2.3.5

The final solution is all the values that make $(x - 10) (x + 3) = 0$ true.

$x = 10, - 3$

Step 4

Set $x - \frac{30}{x} + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - \frac{30}{x} + 1$ equal to $0$ .

$x - \frac{30}{x} + 1 = 0$

Step 4.2

Solve $x - \frac{30}{x} + 1 = 0$ for $x$ .

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Step 4.2.1

Find the LCD of the terms in the equation.

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Step 4.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1, 1$

Step 4.2.1.2

The LCM of one and any expression is the expression.

$x$

Step 4.2.2

Multiply each term in $x - \frac{30}{x} + 1 = 0$ by $x$ to eliminate the fractions.

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Step 4.2.2.1

Multiply each term in $x - \frac{30}{x} + 1 = 0$ by $x$ .

$x \cdot x - \frac{30}{x} x + 1 x = 0 x$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Simplify each term.

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Step 4.2.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} - \frac{30}{x} x + 1 x = 0 x$

Step 4.2.2.2.1.2

Cancel the common factor of $x$ .

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Step 4.2.2.2.1.2.1

Move the leading negative in $- \frac{30}{x}$ into the numerator.

$x^{2} + \frac{- 30}{x} x + 1 x = 0 x$

Step 4.2.2.2.1.2.2

Cancel the common factor.

$x^{2} + \frac{- 30}{x} x + 1 x = 0 x$

Step 4.2.2.2.1.2.3

Rewrite the expression.

$x^{2} - 30 + 1 x = 0 x$

Step 4.2.2.2.1.3

Multiply $x$ by $1$ .

$x^{2} - 30 + x = 0 x$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Multiply $0$ by $x$ .

$x^{2} - 30 + x = 0$

Step 4.2.3

Solve the equation.

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Step 4.2.3.1

Factor $x^{2} - 30 + x$ using the AC method.

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Step 4.2.3.1.1

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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 4.2.3.1.2

Write the factored form using these integers.

$(x - 5) (x + 6) = 0$

Step 4.2.3.2

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

$x - 5 = 0$

$x + 6 = 0$

Step 4.2.3.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.2.3.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.2.3.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.2.3.4

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 4.2.3.4.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 4.2.3.4.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 4.2.3.5

The final solution is all the values that make $(x - 5) (x + 6) = 0$ true.

$x = 5, - 6$

Step 5

The final solution is all the values that make $(x - \frac{30}{x} - 7) (x - \frac{30}{x} + 1) = 0$ true.

$x = 10, - 3, 5, - 6$