Find the Roots/Zeros Using the Rational Roots Test f(x)=15x^4-x^2+25

$f (x) = 15 x^{4} - x^{2} + 25$

Step 1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$15 u^{2} - u + 25 = 0$

$u = x^{2}$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 15$ , $b = - 1$ , and $c = 25$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (15 \cdot 25)}}{2 \cdot 15}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 15 \cdot 25}}{2 \cdot 15}$

Multiply $- 4 \cdot 15 \cdot 25$ .

Multiply $- 4$ by $15$ .

$u = \frac{1 \pm \sqrt{1 - 60 \cdot 25}}{2 \cdot 15}$

Multiply $- 60$ by $25$ .

$u = \frac{1 \pm \sqrt{1 - 1500}}{2 \cdot 15}$

Subtract $1500$ from $1$ .

$u = \frac{1 \pm \sqrt{- 1499}}{2 \cdot 15}$

Rewrite $- 1499$ as $- 1 (1499)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 1499}}{2 \cdot 15}$

Rewrite $\sqrt{- 1 (1499)}$ as $\sqrt{- 1} \cdot \sqrt{1499}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{1499}}{2 \cdot 15}$

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{1 \pm i \sqrt{1499}}{2 \cdot 15}$

Multiply $2$ by $15$ .

$u = \frac{1 \pm i \sqrt{1499}}{30}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 15 \cdot 25}}{2 \cdot 15}$

Multiply $- 4 \cdot 15 \cdot 25$ .

Multiply $- 4$ by $15$ .

$u = \frac{1 \pm \sqrt{1 - 60 \cdot 25}}{2 \cdot 15}$

Multiply $- 60$ by $25$ .

$u = \frac{1 \pm \sqrt{1 - 1500}}{2 \cdot 15}$

Subtract $1500$ from $1$ .

$u = \frac{1 \pm \sqrt{- 1499}}{2 \cdot 15}$

Rewrite $- 1499$ as $- 1 (1499)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 1499}}{2 \cdot 15}$

Rewrite $\sqrt{- 1 (1499)}$ as $\sqrt{- 1} \cdot \sqrt{1499}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{1499}}{2 \cdot 15}$

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{1 \pm i \sqrt{1499}}{2 \cdot 15}$

Multiply $2$ by $15$ .

$u = \frac{1 \pm i \sqrt{1499}}{30}$

Change the $\pm$ to $+$ .

$u = \frac{1 + i \sqrt{1499}}{30}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 15 \cdot 25}}{2 \cdot 15}$

Multiply $- 4 \cdot 15 \cdot 25$ .

Multiply $- 4$ by $15$ .

$u = \frac{1 \pm \sqrt{1 - 60 \cdot 25}}{2 \cdot 15}$

Multiply $- 60$ by $25$ .

$u = \frac{1 \pm \sqrt{1 - 1500}}{2 \cdot 15}$

Subtract $1500$ from $1$ .

$u = \frac{1 \pm \sqrt{- 1499}}{2 \cdot 15}$

Rewrite $- 1499$ as $- 1 (1499)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 1499}}{2 \cdot 15}$

Rewrite $\sqrt{- 1 (1499)}$ as $\sqrt{- 1} \cdot \sqrt{1499}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{1499}}{2 \cdot 15}$

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{1 \pm i \sqrt{1499}}{2 \cdot 15}$

Multiply $2$ by $15$ .

$u = \frac{1 \pm i \sqrt{1499}}{30}$

Change the $\pm$ to $-$ .

$u = \frac{1 - i \sqrt{1499}}{30}$

Step 7

The final answer is the combination of both solutions.

$u = \frac{1 + i \sqrt{1499}}{30}, \frac{1 - i \sqrt{1499}}{30}$

Step 8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{1 + i \sqrt{1499}}{30}$

${(x^{2})}^{1} = \frac{1 - i \sqrt{1499}}{30}$

Step 9

Solve the first equation for $x$ .

$x^{2} = \frac{1 + i \sqrt{1499}}{30}$

Step 10

Solve the equation for $x$ .

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

$x = \pm \sqrt{\frac{1 + i \sqrt{1499}}{30}}$

Simplify $\pm \sqrt{\frac{1 + i \sqrt{1499}}{30}}$ .

Rewrite $\sqrt{\frac{1 + i \sqrt{1499}}{30}}$ as $\frac{\sqrt{1 + i \sqrt{1499}}}{\sqrt{30}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}}}{\sqrt{30}}$

Multiply $\frac{\sqrt{1 + i \sqrt{1499}}}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}}}{\sqrt{30}} \cdot \frac{\sqrt{30}}{\sqrt{30}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{1 + i \sqrt{1499}}}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Raise $\sqrt{30}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{1} \sqrt{30}}$

Raise $\sqrt{30}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{1} {\sqrt{30}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{2}}$

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{{(30^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{30^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{30^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{30^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{30^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 + i \sqrt{1499}} \sqrt{30}}{30}$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}$

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.

$x = \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}$

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

$x = - \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}$

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

$x = \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}, - \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}$

Step 11

Solve the second equation for $x$ .

${(x^{2})}^{1} = \frac{1 - i \sqrt{1499}}{30}$

Step 12

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = \frac{1 - i \sqrt{1499}}{30}$

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

$x = \pm \sqrt{\frac{1 - i \sqrt{1499}}{30}}$

Simplify $\pm \sqrt{\frac{1 - i \sqrt{1499}}{30}}$ .

Rewrite $\sqrt{\frac{1 - i \sqrt{1499}}{30}}$ as $\frac{\sqrt{1 - i \sqrt{1499}}}{\sqrt{30}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}}}{\sqrt{30}}$

Multiply $\frac{\sqrt{1 - i \sqrt{1499}}}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}}}{\sqrt{30}} \cdot \frac{\sqrt{30}}{\sqrt{30}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{1 - i \sqrt{1499}}}{\sqrt{30}}$ by $\frac{\sqrt{30}}{\sqrt{30}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{\sqrt{30} \sqrt{30}}$

Raise $\sqrt{30}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{1} \sqrt{30}}$

Raise $\sqrt{30}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{1} {\sqrt{30}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{{\sqrt{30}}^{2}}$

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{{(30^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{30^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{30^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{30^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{30^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 - i \sqrt{1499}} \sqrt{30}}{30}$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}$

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.

$x = \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}$

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

$x = - \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}$

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

$x = \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}, - \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}$

Step 13

The solution to $15 x^{4} - x^{2} + 25 = 0$ is $x = \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}, - \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}, \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}, - \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}$ .

$x = \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}, - \frac{\sqrt{(1 + i \sqrt{1499}) \cdot 30}}{30}, \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}, - \frac{\sqrt{(1 - i \sqrt{1499}) \cdot 30}}{30}$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=15x^4-x^2+25

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