Solve by Factoring 3x^4-31x^2+28=0

$3 x^{4} - 31 x^{2} + 28 = 0$

Step 1

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

$3 {(x^{2})}^{2} - 31 x^{2} + 28 = 0$

Step 2

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

$3 u^{2} - 31 u + 28 = 0$

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 $a \cdot c = 3 \cdot 28 = 84$ and whose sum is $b = - 31$ .

Factor $- 31$ out of $- 31 u$ .

$3 u^{2} - 31 u + 28 = 0$

Rewrite $- 31$ as $- 3$ plus $- 28$

$3 u^{2} + (- 3 - 28) u + 28 = 0$

Apply the distributive property.

$3 u^{2} - 3 u - 28 u + 28 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 u^{2} - 3 u) - 28 u + 28 = 0$

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

$3 u (u - 1) - 28 (u - 1) = 0$

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

$(u - 1) (3 u - 28) = 0$

Step 4

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

$(x^{2} - 1) (3 x^{2} - 28) = 0$

Step 5

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

$(x^{2} - 1^{2}) (3 x^{2} - 28) = 0$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$(x + 1) (x - 1) (3 x^{2} - 28) = 0$

Step 7

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

$x + 1 = 0$

$x - 1 = 0$

$3 x^{2} - 28 = 0$

Step 8

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 10

Set $3 x^{2} - 28$ equal to $0$ and solve for $x$ .

Set $3 x^{2} - 28$ equal to $0$ .

$3 x^{2} - 28 = 0$

Solve $3 x^{2} - 28 = 0$ for $x$ .

Add $28$ to both sides of the equation.

$3 x^{2} = 28$

Divide each term in $3 x^{2} = 28$ by $3$ and simplify.

Divide each term in $3 x^{2} = 28$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{28}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{28}{3}$

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

$x^{2} = \frac{28}{3}$

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

$x = \pm \sqrt{\frac{28}{3}}$

Simplify $\pm \sqrt{\frac{28}{3}}$ .

Rewrite $\sqrt{\frac{28}{3}}$ as $\frac{\sqrt{28}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{28}}{\sqrt{3}}$

Simplify the numerator.

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$x = \pm \frac{\sqrt{4 (7)}}{\sqrt{3}}$

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

$x = \pm \frac{\sqrt{2^{2} \cdot 7}}{\sqrt{3}}$

Pull terms out from under the radical.

$x = \pm \frac{2 \sqrt{7}}{\sqrt{3}}$

Multiply $\frac{2 \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2 \sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{2 \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{7} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{2 \sqrt{3 \cdot 7}}{3}$

Multiply $3$ by $7$ .

$x = \pm \frac{2 \sqrt{21}}{3}$

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.

$x = \frac{2 \sqrt{21}}{3}$

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

$x = - \frac{2 \sqrt{21}}{3}$

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

$x = \frac{2 \sqrt{21}}{3}, - \frac{2 \sqrt{21}}{3}$

Step 11

The final solution is all the values that make $(x + 1) (x - 1) (3 x^{2} - 28) = 0$ true.

$x = - 1, 1, \frac{2 \sqrt{21}}{3}, - \frac{2 \sqrt{21}}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - 1, 1, \frac{2 \sqrt{21}}{3}, - \frac{2 \sqrt{21}}{3}$

Decimal Form:

$x = - 1, 1, 3.05505046 \dots, - 3.05505046 \dots$

Solve by Factoring 3x^4-31x^2+28=0

Related Questions

Support LCMGCF.com by disabling your adblocker.