Algebra Examples

$\frac{3 x^{2} + 19 x + 20}{9 x^{2} - 16} \cdot \frac{3 x^{2} + 8 x - 16}{x^{2} + x - 20}$

Step 1

Set the denominator in $\frac{3 x^{2} + 19 x + 20}{9 x^{2} - 16}$ equal to $0$ to find where the expression is undefined.

$9 x^{2} - 16 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $16$ to both sides of the equation.

$9 x^{2} = 16$

Step 2.2

Divide each term in $9 x^{2} = 16$ by $9$ and simplify.

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Step 2.2.1

Divide each term in $9 x^{2} = 16$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{16}{9}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $9$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{16}{9}$

Step 2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{16}{9}$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{16}{9}}$

Step 2.4

Simplify $\pm \sqrt{\frac{16}{9}}$ .

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Step 2.4.1

Rewrite $\sqrt{\frac{16}{9}}$ as $\frac{\sqrt{16}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{16}}{\sqrt{9}}$

Step 2.4.2

Simplify the numerator.

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Step 2.4.2.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \frac{\sqrt{4^{2}}}{\sqrt{9}}$

Step 2.4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{4}{\sqrt{9}}$

Step 2.4.3

Simplify the denominator.

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Step 2.4.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{4}{\sqrt{3^{2}}}$

Step 2.4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{4}{3}$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{4}{3}$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{4}{3}$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{4}{3}, - \frac{4}{3}$

Step 3

Set the denominator in $\frac{3 x^{2} + 8 x - 16}{x^{2} + x - 20}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x - 20 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Factor $x^{2} + x - 20$ using the AC method.

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Step 4.1.1

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 4.1.2

Write the factored form using these integers.

$(x - 4) (x + 5) = 0$

Step 4.2

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

$x - 4 = 0$

$x + 5 = 0$

Step 4.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.4

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

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Step 4.4.1

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

$x + 5 = 0$

Step 4.4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.5

The final solution is all the values that make $(x - 4) (x + 5) = 0$ true.

$x = 4, - 5$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 5, x = - \frac{4}{3}, x = \frac{4}{3}, x = 4$

Step 6