Solve by Factoring 9x^4-16=0

$9 x^{4} - 16 = 0$

Step 1

Rewrite $9 x^{4}$ as ${(3 x^{2})}^{2}$ .

${(3 x^{2})}^{2} - 16 = 0$

Step 2

Rewrite $16$ as $4^{2}$ .

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x^{2}$ and $b = 4$ .

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

Step 4

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

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

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

Step 5

Set $3 x^{2} + 4$ equal to $0$ and solve for $x$ .

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

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

Solve $3 x^{2} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$3 x^{2} = - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

$x = \pm \sqrt{- \frac{4}{3}}$

Simplify $\pm \sqrt{- \frac{4}{3}}$ .

Rewrite $- \frac{4}{3}$ as $i^{2} \frac{2^{2}}{3}$ .

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

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

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

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

Pull terms out from under the radical.

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

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

$x = \pm i \sqrt{\frac{4}{3}}$

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{4}}{\sqrt{3}}$

Simplify the numerator.

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Combine $i$ and $\frac{2 \sqrt{3}}{3}$ .

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

Move $2$ to the left of $i$ .

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

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.

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

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

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

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

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

Step 6

Set $3 x^{2} - 4$ equal to $0$ and solve for $x$ .

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

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

Solve $3 x^{2} - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

$3 x^{2} = 4$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

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

$x = \pm \sqrt{\frac{4}{3}}$

Simplify $\pm \sqrt{\frac{4}{3}}$ .

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{3}}$

Simplify the numerator.

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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.

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

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

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

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

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

Step 7

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

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

Solve by Factoring 9x^4-16=0

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