Solve by Factoring 9x^4-18x^2+5=0

$9 x^{4} - 18 x^{2} + 5 = 0$

Step 1

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

$9 {(x^{2})}^{2} - 18 x^{2} + 5 = 0$

Step 2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$9 u^{2} - 18 u + 5 = 0$

Step 3

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 $a \cdot c = 9 \cdot 5 = 45$ and whose sum is $b = - 18$ .

Factor $- 18$ out of $- 18 u$ .

$9 u^{2} - 18 u + 5 = 0$

Rewrite $- 18$ as $- 3$ plus $- 15$

$9 u^{2} + (- 3 - 15) u + 5 = 0$

Apply the distributive property.

$9 u^{2} - 3 u - 15 u + 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 u^{2} - 3 u) - 15 u + 5 = 0$

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

$3 u (3 u - 1) - 5 (3 u - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

$(3 u - 1) (3 u - 5) = 0$

Step 4

Replace all occurrences of $u$ with $x^{2}$ .

$(3 x^{2} - 1) (3 x^{2} - 5) = 0$

Step 5

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

$3 x^{2} - 1 = 0$

$3 x^{2} - 5 = 0$

Step 6

Set $3 x^{2} - 1$ equal to $0$ and solve for $x$ .

Set $3 x^{2} - 1$ equal to $0$ .

$3 x^{2} - 1 = 0$

Solve $3 x^{2} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x^{2} = 1$

Divide each term in $3 x^{2} = 1$ by $3$ and simplify.

Divide each term in $3 x^{2} = 1$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

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

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

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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.

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

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

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

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

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

Step 7

Set $3 x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $3 x^{2} - 5$ equal to $0$ .

$3 x^{2} - 5 = 0$

Solve $3 x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$3 x^{2} = 5$

Divide each term in $3 x^{2} = 5$ by $3$ and simplify.

Divide each term in $3 x^{2} = 5$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{5}{3}$

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

$x^{2} = \frac{5}{3}$

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

$x = \pm \sqrt{\frac{5}{3}}$

Simplify $\pm \sqrt{\frac{5}{3}}$ .

Rewrite $\sqrt{\frac{5}{3}}$ as $\frac{\sqrt{5}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{3}}$

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$x = \pm \frac{\sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{5} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{5} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{5 \cdot 3}}{3}$

Multiply $5$ by $3$ .

$x = \pm \frac{\sqrt{15}}{3}$

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.

$x = \frac{\sqrt{15}}{3}$

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

$x = - \frac{\sqrt{15}}{3}$

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

$x = \frac{\sqrt{15}}{3}, - \frac{\sqrt{15}}{3}$

Step 8

The final solution is all the values that make $(3 x^{2} - 1) (3 x^{2} - 5) = 0$ true.

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}, \frac{\sqrt{15}}{3}, - \frac{\sqrt{15}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}, \frac{\sqrt{15}}{3}, - \frac{\sqrt{15}}{3}$

Decimal Form:

$x = 0.57735026 \dots, - 0.57735026 \dots, 1.29099444 \dots, - 1.29099444 \dots$

Solve by Factoring 9x^4-18x^2+5=0

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