Algebra Examples

$9 x^{4} = 25 x^{2} - 16$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $25 x^{2}$ from both sides of the equation.

$9 x^{4} - 25 x^{2} = - 16$

Step 1.2

Add $16$ to both sides of the equation.

$9 x^{4} - 25 x^{2} + 16 = 0$

Step 2

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

$9 {(x^{2})}^{2} - 25 x^{2} + 16 = 0$

Step 3

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

$9 u^{2} - 25 u + 16 = 0$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 9 \cdot 16 = 144$ and whose sum is $b = - 25$ .

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Step 4.1.1

Factor $- 25$ out of $- 25 u$ .

$9 u^{2} - 25 u + 16 = 0$

Step 4.1.2

Rewrite $- 25$ as $- 9$ plus $- 16$

$9 u^{2} + (- 9 - 16) u + 16 = 0$

Step 4.1.3

Apply the distributive property.

$9 u^{2} - 9 u - 16 u + 16 = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$(9 u^{2} - 9 u) - 16 u + 16 = 0$

Step 4.2.2

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

$9 u (u - 1) - 16 (u - 1) = 0$

Step 4.3

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

$(u - 1) (9 u - 16) = 0$

Step 5

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

$(x^{2} - 1) (9 x^{2} - 16) = 0$

Step 6

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

$(x^{2} - 1^{2}) (9 x^{2} - 16) = 0$

Step 7

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

$(x + 1) (x - 1) (9 x^{2} - 16) = 0$

Step 8

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

$(x + 1) (x - 1) ({(3 x)}^{2} - 16) = 0$

Step 9

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

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

Step 10

Factor.

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Step 10.1

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

$(x + 1) (x - 1) ((3 x + 4) (3 x - 4)) = 0$

Step 10.2

Remove unnecessary parentheses.

$(x + 1) (x - 1) (3 x + 4) (3 x - 4) = 0$

Step 11

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

$x + 1 = 0$

$x - 1 = 0$

$3 x + 4 = 0$

$3 x - 4 = 0$

Step 12

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

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Step 12.1

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

$x + 1 = 0$

Step 12.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 13

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

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Step 13.1

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

$x - 1 = 0$

Step 13.2

Add $1$ to both sides of the equation.

$x = 1$

Step 14

Set $3 x + 4$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Step 14.2

Solve $3 x + 4 = 0$ for $x$ .

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Step 14.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 14.2.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

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Step 14.2.2.1

Divide each term in $3 x = - 4$ by $3$ .

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

Step 14.2.2.2

Simplify the left side.

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Step 14.2.2.2.1

Cancel the common factor of $3$ .

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Step 14.2.2.2.1.1

Cancel the common factor.

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

Step 14.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 14.2.2.3

Simplify the right side.

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Step 14.2.2.3.1

Move the negative in front of the fraction.

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

Step 15

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 15.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 15.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 15.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 15.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 15.2.2.1

Divide each term in $3 x = 4$ by $3$ .

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

Step 15.2.2.2

Simplify the left side.

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Step 15.2.2.2.1

Cancel the common factor of $3$ .

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Step 15.2.2.2.1.1

Cancel the common factor.

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

Step 15.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 16

The final solution is all the values that make $(x + 1) (x - 1) (3 x + 4) (3 x - 4) = 0$ true.

$x = - 1, 1, - \frac{4}{3}, \frac{4}{3}$