Find All Complex Solutions x^4-53x^2+196=0

$x^{4} - 53 x^{2} + 196 = 0$

Step 1

Factor the left side of the equation.

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

${(x^{2})}^{2} - 53 x^{2} + 196 = 0$

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

$u^{2} - 53 u + 196 = 0$

Factor $u^{2} - 53 u + 196$ 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 $196$ and whose sum is $- 53$ .

$- 49, - 4$

Write the factored form using these integers.

$(u - 49) (u - 4) = 0$

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

$(x^{2} - 49) (x^{2} - 4) = 0$

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

$(x^{2} - 7^{2}) (x^{2} - 4) = 0$

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 $b = 7$ .

$(x + 7) (x - 7) (x^{2} - 4) = 0$

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

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

Factor.

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 $b = 2$ .

$(x + 7) (x - 7) ((x + 2) (x - 2)) = 0$

Remove unnecessary parentheses.

$(x + 7) (x - 7) (x + 2) (x - 2) = 0$

Step 2

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

$x + 7 = 0$

$x - 7 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 3

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

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

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 4

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

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 5

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $(x + 7) (x - 7) (x + 2) (x - 2) = 0$ true.

$x = - 7, 7, - 2, 2$

Find All Complex Solutions x^4-53x^2+196=0

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