Solve by Factoring 36x^4=85x^2-49

$36 x^{4} = 85 x^{2} - 49$

Step 1

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

Subtract $85 x^{2}$ from both sides of the equation.

$36 x^{4} - 85 x^{2} = - 49$

Add $49$ to both sides of the equation.

$36 x^{4} - 85 x^{2} + 49 = 0$

Step 2

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

$36 {(x^{2})}^{2} - 85 x^{2} + 49 = 0$

Step 3

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

$36 u^{2} - 85 u + 49 = 0$

Step 4

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 $a \cdot c = 36 \cdot 49 = 1764$ and whose sum is $b = - 85$ .

Factor $- 85$ out of $- 85 u$ .

$36 u^{2} - 85 u + 49 = 0$

Rewrite $- 85$ as $- 36$ plus $- 49$

$36 u^{2} + (- 36 - 49) u + 49 = 0$

Apply the distributive property.

$36 u^{2} - 36 u - 49 u + 49 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(36 u^{2} - 36 u) - 49 u + 49 = 0$

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

$36 u (u - 1) - 49 (u - 1) = 0$

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

$(u - 1) (36 u - 49) = 0$

Step 5

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

$(x^{2} - 1) (36 x^{2} - 49) = 0$

Step 6

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

$(x^{2} - 1^{2}) (36 x^{2} - 49) = 0$

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 $b = 1$ .

$(x + 1) (x - 1) (36 x^{2} - 49) = 0$

Step 8

Rewrite $36 x^{2}$ as ${(6 x)}^{2}$ .

$(x + 1) (x - 1) ({(6 x)}^{2} - 49) = 0$

Step 9

Rewrite $49$ as $7^{2}$ .

$(x + 1) (x - 1) ({(6 x)}^{2} - 7^{2}) = 0$

Step 10

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6 x$ and $b = 7$ .

$(x + 1) (x - 1) ((6 x + 7) (6 x - 7)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 1) (6 x + 7) (6 x - 7) = 0$

Step 11

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

$x + 1 = 0$

$x - 1 = 0$

$6 x + 7 = 0$

$6 x - 7 = 0$

Step 12

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 13

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 14

Set $6 x + 7$ equal to $0$ and solve for $x$ .

Set $6 x + 7$ equal to $0$ .

$6 x + 7 = 0$

Solve $6 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$6 x = - 7$

Divide each term in $6 x = - 7$ by $6$ and simplify.

Divide each term in $6 x = - 7$ by $6$ .

$\frac{6 x}{6} = \frac{- 7}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 7}{6}$

Divide $x$ by $1$ .

$x = \frac{- 7}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{6}$

Step 15

Set $6 x - 7$ equal to $0$ and solve for $x$ .

Set $6 x - 7$ equal to $0$ .

$6 x - 7 = 0$

Solve $6 x - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$6 x = 7$

Divide each term in $6 x = 7$ by $6$ and simplify.

Divide each term in $6 x = 7$ by $6$ .

$\frac{6 x}{6} = \frac{7}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{7}{6}$

Divide $x$ by $1$ .

$x = \frac{7}{6}$

Step 16

The final solution is all the values that make $(x + 1) (x - 1) (6 x + 7) (6 x - 7) = 0$ true.

$x = - 1, 1, - \frac{7}{6}, \frac{7}{6}$

Solve by Factoring 36x^4=85x^2-49

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