Find the Excluded Values (32x^4-50)/(4x^3-12x^2-5x+15)

$\frac{32 x^{4} - 50}{4 x^{3} - 12 x^{2} - 5 x + 15}$

Set the denominator in $\frac{32 x^{4} - 50}{4 x^{3} - 12 x^{2} - 5 x + 15}$ equal to $0$ to find where the expression is undefined.

$4 x^{3} - 12 x^{2} - 5 x + 15 = 0$

Solve for $x$ .

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{3} - 12 x^{2}) - 5 x + 15 = 0$

Factor out the greatest common factor (GCF) from each group.

$4 x^{2} (x - 3) - 5 (x - 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$(x - 3) (4 x^{2} - 5) = 0$

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

$x - 3 = 0$

$4 x^{2} - 5 = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $4 x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $4 x^{2} - 5$ equal to $0$ .

$4 x^{2} - 5 = 0$

Solve $4 x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$4 x^{2} = 5$

Divide each term in $4 x^{2} = 5$ by $4$ and simplify.

Divide each term in $4 x^{2} = 5$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{5}{4}$

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

$x^{2} = \frac{5}{4}$

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

$x = \pm \sqrt{\frac{5}{4}}$

Simplify $\pm \sqrt{\frac{5}{4}}$ .

Rewrite $\sqrt{\frac{5}{4}}$ as $\frac{\sqrt{5}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{4}}$

Simplify the denominator.

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

$x = \pm \frac{\sqrt{5}}{\sqrt{2^{2}}}$

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

$x = \pm \frac{\sqrt{5}}{2}$

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

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

$x = \frac{\sqrt{5}}{2}$

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

$x = - \frac{\sqrt{5}}{2}$

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

$x = \frac{\sqrt{5}}{2}, - \frac{\sqrt{5}}{2}$

The final solution is all the values that make $(x - 3) (4 x^{2} - 5) = 0$ true.

$x = 3, \frac{\sqrt{5}}{2}, - \frac{\sqrt{5}}{2}$

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 $0$ .

$x = - \frac{\sqrt{5}}{2}, x = \frac{\sqrt{5}}{2}, x = 3$

Find the Excluded Values (32x^4-50)/(4x^3-12x^2-5x+15)

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