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

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

Step 1

Set $- 2 x^{4} + 3$ equal to $0$ .

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

Step 2

Solve for $x$ .

Subtract $3$ from both sides of the equation.

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

Divide each term in $- 2 x^{4} = - 3$ by $- 2$ and simplify.

Divide each term in $- 2 x^{4} = - 3$ by $- 2$ .

$\frac{- 2 x^{4}}{- 2} = \frac{- 3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x^{4}}{- 2} = \frac{- 3}{- 2}$

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 3}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

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

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

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

Simplify $\pm \sqrt[4]{\frac{3}{2}}$ .

Rewrite $\sqrt[4]{\frac{3}{2}}$ as $\frac{\sqrt[4]{3}}{\sqrt[4]{2}}$ .

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

Multiply $\frac{\sqrt[4]{3}}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{3}}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

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

Raise $\sqrt[4]{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt[4]{3} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1} {\sqrt[4]{2}}^{3}}$

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

$x = \pm \frac{\sqrt[4]{3} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1 + 3}}$

Add $1$ and $3$ .

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

Rewrite ${\sqrt[4]{2}}^{4}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2}$ as $2^{\frac{1}{4}}$ .

$x = \pm \frac{\sqrt[4]{3} {\sqrt[4]{2}}^{3}}{{(2^{\frac{1}{4}})}^{4}}$

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

$x = \pm \frac{\sqrt[4]{3} {\sqrt[4]{2}}^{3}}{2^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$x = \pm \frac{\sqrt[4]{3} {\sqrt[4]{2}}^{3}}{2^{1}}$

Evaluate the exponent.

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

Simplify the numerator.

Rewrite ${\sqrt[4]{2}}^{3}$ as ${(2^{3})}^{\frac{1}{4}}$ .

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

Raise $2$ to the power of $3$ .

$x = \pm \frac{\sqrt[4]{3} \sqrt[4]{8}}{2}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt[4]{3 \cdot 8}}{2}$

Multiply $3$ by $8$ .

$x = \pm \frac{\sqrt[4]{24}}{2}$

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.

$x = \frac{\sqrt[4]{24}}{2}$

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

$x = - \frac{\sqrt[4]{24}}{2}$

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

$x = \frac{\sqrt[4]{24}}{2}, - \frac{\sqrt[4]{24}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[4]{24}}{2}, - \frac{\sqrt[4]{24}}{2}$

Decimal Form:

$x = 1.10668191 \dots, - 1.10668191 \dots$

Step 4

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

Related Questions

Support LCMGCF.com by disabling your adblocker.