Solve by Factoring y^4-37y^2+36=0

$y^{4} - 37 y^{2} + 36 = 0$

Step 1

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

${(y^{2})}^{2} - 37 y^{2} + 36 = 0$

Step 2

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

$u^{2} - 37 u + 36 = 0$

Step 3

Factor $u^{2} - 37 u + 36$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $36$ and whose sum is $- 37$ .

$- 36, - 1$

Write the factored form using these integers.

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

Step 4

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

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

Step 5

Rewrite $36$ as $6^{2}$ .

$(y^{2} - 6^{2}) (y^{2} - 1) = 0$

Step 6

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

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

Step 7

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

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

Step 8

Factor.

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

$(y + 6) (y - 6) ((y + 1) (y - 1)) = 0$

Remove unnecessary parentheses.

$(y + 6) (y - 6) (y + 1) (y - 1) = 0$

Step 9

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

$y + 6 = 0$

$y - 6 = 0$

$y + 1 = 0$

$y - 1 = 0$

Step 10

Set $y + 6$ equal to $0$ and solve for $y$ .

Set $y + 6$ equal to $0$ .

$y + 6 = 0$

Subtract $6$ from both sides of the equation.

$y = - 6$

Step 11

Set $y - 6$ equal to $0$ and solve for $y$ .

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Add $6$ to both sides of the equation.

$y = 6$

Step 12

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

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

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 13

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

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

$y - 1 = 0$

Add $1$ to both sides of the equation.

$y = 1$

Step 14

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

$y = - 6, 6, - 1, 1$

Solve by Factoring y^4-37y^2+36=0

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