Algebra Examples

$9 a^{4} - 37 a^{2} + 4 = 0$

Step 1

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

$9 {(a^{2})}^{2} - 37 a^{2} + 4 = 0$

Step 2

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

$9 u^{2} - 37 u + 4 = 0$

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 $a \cdot c = 9 \cdot 4 = 36$ and whose sum is $b = - 37$ .

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Step 3.1.1

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

$9 u^{2} - 37 u + 4 = 0$

Step 3.1.2

Rewrite $- 37$ as $- 1$ plus $- 36$

$9 u^{2} + (- 1 - 36) u + 4 = 0$

Step 3.1.3

Apply the distributive property.

$9 u^{2} - 1 u - 36 u + 4 = 0$

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.

$(9 u^{2} - 1 u) - 36 u + 4 = 0$

Step 3.2.2

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

$u (9 u - 1) - 4 (9 u - 1) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $9 u - 1$ .

$(9 u - 1) (u - 4) = 0$

Step 4

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

$(9 a^{2} - 1) (a^{2} - 4) = 0$

Step 5

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

$({(3 a)}^{2} - 1) (a^{2} - 4) = 0$

Step 6

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

$({(3 a)}^{2} - 1^{2}) (a^{2} - 4) = 0$

Step 7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 a$ and $b = 1$ .

$(3 a + 1) (3 a - 1) (a^{2} - 4) = 0$

Step 8

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

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

Step 9

Factor.

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Step 9.1

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

$(3 a + 1) (3 a - 1) ((a + 2) (a - 2)) = 0$

Step 9.2

Remove unnecessary parentheses.

$(3 a + 1) (3 a - 1) (a + 2) (a - 2) = 0$

Step 10

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

$3 a + 1 = 0$

$3 a - 1 = 0$

$a + 2 = 0$

$a - 2 = 0$

Step 11

Set $3 a + 1$ equal to $0$ and solve for $a$ .

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Step 11.1

Set $3 a + 1$ equal to $0$ .

$3 a + 1 = 0$

Step 11.2

Solve $3 a + 1 = 0$ for $a$ .

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Step 11.2.1

Subtract $1$ from both sides of the equation.

$3 a = - 1$

Step 11.2.2

Divide each term in $3 a = - 1$ by $3$ and simplify.

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Step 11.2.2.1

Divide each term in $3 a = - 1$ by $3$ .

$\frac{3 a}{3} = \frac{- 1}{3}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $3$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{- 1}{3}$

Step 11.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 1}{3}$

Step 11.2.2.3

Simplify the right side.

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Step 11.2.2.3.1

Move the negative in front of the fraction.

$a = - \frac{1}{3}$

Step 12

Set $3 a - 1$ equal to $0$ and solve for $a$ .

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Step 12.1

Set $3 a - 1$ equal to $0$ .

$3 a - 1 = 0$

Step 12.2

Solve $3 a - 1 = 0$ for $a$ .

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Step 12.2.1

Add $1$ to both sides of the equation.

$3 a = 1$

Step 12.2.2

Divide each term in $3 a = 1$ by $3$ and simplify.

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Step 12.2.2.1

Divide each term in $3 a = 1$ by $3$ .

$\frac{3 a}{3} = \frac{1}{3}$

Step 12.2.2.2

Simplify the left side.

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Step 12.2.2.2.1

Cancel the common factor of $3$ .

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Step 12.2.2.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{1}{3}$

Step 12.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{1}{3}$

Step 13

Set $a + 2$ equal to $0$ and solve for $a$ .

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Step 13.1

Set $a + 2$ equal to $0$ .

$a + 2 = 0$

Step 13.2

Subtract $2$ from both sides of the equation.

$a = - 2$

Step 14

Set $a - 2$ equal to $0$ and solve for $a$ .

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Step 14.1

Set $a - 2$ equal to $0$ .

$a - 2 = 0$

Step 14.2

Add $2$ to both sides of the equation.

$a = 2$

Step 15

The final solution is all the values that make $(3 a + 1) (3 a - 1) (a + 2) (a - 2) = 0$ true.

$a = - \frac{1}{3}, \frac{1}{3}, - 2, 2$