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

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

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

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

$u = x^{2}$

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

$3 u^{2} + 1 u - 2 = 0$

Rewrite $1$ as $- 2$ plus $3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$u (3 u - 2) + 1 (3 u - 2) = 0$

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

$(3 u - 2) (u + 1) = 0$

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

$3 u - 2 = 0$

$u + 1 = 0$

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 $u$ .

Add $2$ to 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 $1$ .

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

Set $u + 1$ equal to $0$ and solve for $u$ .

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Subtract $1$ from both sides of the equation.

$u = - 1$

The final solution is all the values that make $(3 u - 2) (u + 1) = 0$ true.

$u = \frac{2}{3}, - 1$

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

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

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

Solve the first equation for $x$ .

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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.

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

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

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

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

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

Solve the second equation for $x$ .

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

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = - 1$

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

$x = \pm \sqrt{- 1}$

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

$x = \pm i$

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.

$x = i$

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

$x = - i$

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

$x = i, - i$

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

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

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

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