Algebra Examples

$11 x^{4} - 37 x^{2} + 26 = 0$

Step 1

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

$11 {(x^{2})}^{2} - 37 x^{2} + 26 = 0$

Step 2

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

$11 u^{2} - 37 u + 26 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 11 \cdot 26 = 286$ and whose sum is $b = - 37$ .

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Step 3.1.1

Factor $- 37$ out of $- 37 u$ .

$11 u^{2} - 37 u + 26 = 0$

Step 3.1.2

Rewrite $- 37$ as $- 11$ plus $- 26$

$11 u^{2} + (- 11 - 26) u + 26 = 0$

Step 3.1.3

Apply the distributive property.

$11 u^{2} - 11 u - 26 u + 26 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(11 u^{2} - 11 u) - 26 u + 26 = 0$

Step 3.2.2

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

$11 u (u - 1) - 26 (u - 1) = 0$

Step 3.3

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

$(u - 1) (11 u - 26) = 0$

Step 4

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

$(x^{2} - 1) (11 x^{2} - 26) = 0$

Step 5

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

$(x^{2} - 1^{2}) (11 x^{2} - 26) = 0$

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 $b = 1$ .

$(x + 1) (x - 1) (11 x^{2} - 26) = 0$

Step 7

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

$x + 1 = 0$

$x - 1 = 0$

$11 x^{2} - 26 = 0$

Step 8

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

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Step 8.1

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

$x + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

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

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Step 9.1

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

$x - 1 = 0$

Step 9.2

Add $1$ to both sides of the equation.

$x = 1$

Step 10

Set $11 x^{2} - 26$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $11 x^{2} - 26$ equal to $0$ .

$11 x^{2} - 26 = 0$

Step 10.2

Solve $11 x^{2} - 26 = 0$ for $x$ .

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Step 10.2.1

Add $26$ to both sides of the equation.

$11 x^{2} = 26$

Step 10.2.2

Divide each term in $11 x^{2} = 26$ by $11$ and simplify.

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Step 10.2.2.1

Divide each term in $11 x^{2} = 26$ by $11$ .

$\frac{11 x^{2}}{11} = \frac{26}{11}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $11$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{11 x^{2}}{11} = \frac{26}{11}$

Step 10.2.2.2.1.2

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

$x^{2} = \frac{26}{11}$

Step 10.2.3

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

$x = \pm \sqrt{\frac{26}{11}}$

Step 10.2.4

Simplify $\pm \sqrt{\frac{26}{11}}$ .

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Step 10.2.4.1

Rewrite $\sqrt{\frac{26}{11}}$ as $\frac{\sqrt{26}}{\sqrt{11}}$ .

$x = \pm \frac{\sqrt{26}}{\sqrt{11}}$

Step 10.2.4.2

Multiply $\frac{\sqrt{26}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \pm \frac{\sqrt{26}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 10.2.4.3

Combine and simplify the denominator.

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Step 10.2.4.3.1

Multiply $\frac{\sqrt{26}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \pm \frac{\sqrt{26} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 10.2.4.3.2

Raise $\sqrt{11}$ to the power of $1$ .

$x = \pm \frac{\sqrt{26} \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Step 10.2.4.3.3

Raise $\sqrt{11}$ to the power of $1$ .

$x = \pm \frac{\sqrt{26} \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

Step 10.2.4.3.4

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

$x = \pm \frac{\sqrt{26} \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 10.2.4.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{26} \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 10.2.4.3.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 10.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{26} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 10.2.4.3.6.2

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

$x = \pm \frac{\sqrt{26} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 10.2.4.3.6.3

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

$x = \pm \frac{\sqrt{26} \sqrt{11}}{11^{\frac{2}{2}}}$

Step 10.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 10.2.4.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{26} \sqrt{11}}{11^{\frac{2}{2}}}$

Step 10.2.4.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{26} \sqrt{11}}{11^{1}}$

Step 10.2.4.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{26} \sqrt{11}}{11}$

Step 10.2.4.4

Simplify the numerator.

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Step 10.2.4.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{26 \cdot 11}}{11}$

Step 10.2.4.4.2

Multiply $26$ by $11$ .

$x = \pm \frac{\sqrt{286}}{11}$

Step 10.2.5

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

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Step 10.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{286}}{11}$

Step 10.2.5.2

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

$x = - \frac{\sqrt{286}}{11}$

Step 10.2.5.3

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

$x = \frac{\sqrt{286}}{11}, - \frac{\sqrt{286}}{11}$

Step 11

The final solution is all the values that make $(x + 1) (x - 1) (11 x^{2} - 26) = 0$ true.

$x = - 1, 1, \frac{\sqrt{286}}{11}, - \frac{\sqrt{286}}{11}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - 1, 1, \frac{\sqrt{286}}{11}, - \frac{\sqrt{286}}{11}$

Decimal Form:

$x = - 1, 1, 1.53741222 \dots, - 1.53741222 \dots$