Find the Roots (Zeros) 3x^4-16x^2-12=0

$3 x^{4} - 16 x^{2} - 12 = 0$

Step 1

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

$3 u^{2} - 16 u - 12 = 0$

$u = x^{2}$

Step 2

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 12 = - 36$ and whose sum is $b = - 16$ .

Factor $- 16$ out of $- 16 u$ .

$3 u^{2} - 16 u - 12 = 0$

Rewrite $- 16$ as $2$ plus $- 18$

$3 u^{2} + (2 - 18) u - 12 = 0$

Apply the distributive property.

$3 u^{2} + 2 u - 18 u - 12 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 u^{2} + 2 u) - 18 u - 12 = 0$

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

$u (3 u + 2) - 6 (3 u + 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 u + 2$ .

$(3 u + 2) (u - 6) = 0$

Step 3

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

$3 u + 2 = 0$

$u - 6 = 0$

Step 4

Set $3 u + 2$ equal to $0$ and solve for $u$ .

Set $3 u + 2$ equal to $0$ .

$3 u + 2 = 0$

Solve $3 u + 2 = 0$ for $u$ .

Subtract $2$ from both sides of the equation.

$3 u = - 2$

Divide each term in $3 u = - 2$ by $3$ and simplify.

Divide each term in $3 u = - 2$ by $3$ .

$\frac{3 u}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 u}{3} = \frac{- 2}{3}$

Divide $u$ by $1$ .

$u = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$u = - \frac{2}{3}$

Step 5

Set $u - 6$ equal to $0$ and solve for $u$ .

Set $u - 6$ equal to $0$ .

$u - 6 = 0$

Add $6$ to both sides of the equation.

$u = 6$

Step 6

The final solution is all the values that make $(3 u + 2) (u - 6) = 0$ true.

$u = - \frac{2}{3}, 6$

Step 7

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

$x^{2} = - \frac{2}{3}$

${(x^{2})}^{1} = 6$

Step 8

Solve the first equation for $x$ .

$x^{2} = - \frac{2}{3}$

Step 9

Solve the equation for $x$ .

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

$x = \pm \sqrt{- \frac{2}{3}}$

Simplify $\pm \sqrt{- \frac{2}{3}}$ .

Rewrite $- 1$ as $i^{2}$ .

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

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{2}{3}}$

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{3}}$

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{2 \cdot 3}}{3}$

Multiply $2$ by $3$ .

$x = \pm i \frac{\sqrt{6}}{3}$

Combine $i$ and $\frac{\sqrt{6}}{3}$ .

$x = \pm \frac{i \sqrt{6}}{3}$

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.

$x = \frac{i \sqrt{6}}{3}$

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

$x = - \frac{i \sqrt{6}}{3}$

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

$x = \frac{i \sqrt{6}}{3}, - \frac{i \sqrt{6}}{3}$

Step 10

Solve the second equation for $x$ .

${(x^{2})}^{1} = 6$

Step 11

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = 6$

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

$x = \pm \sqrt{6}$

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.

$x = \sqrt{6}$

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

$x = - \sqrt{6}$

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

$x = \sqrt{6}, - \sqrt{6}$

Step 12

The solution to $3 x^{4} - 16 x^{2} - 12 = 0$ is $x = \frac{i \sqrt{6}}{3}, - \frac{i \sqrt{6}}{3}, \sqrt{6}, - \sqrt{6}$ .

$x = \frac{i \sqrt{6}}{3}, - \frac{i \sqrt{6}}{3}, \sqrt{6}, - \sqrt{6}$

Step 13

Find the Roots (Zeros) 3x^4-16x^2-12=0

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