Find the Roots (Zeros) x^4-10x^2+9=0

$x^{4} - 10 x^{2} + 9 = 0$

Factor the left side of the equation.

Tap for more steps…

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

${(x^{2})}^{2} - 10 x^{2} + 9$

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

$u^{2} - 10 u + 9$

Factor $u^{2} - 10 u + 9$ using the AC method.

Tap for more steps…

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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$(u - 9) (u - 1)$

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

$(x^{2} - 9) (x^{2} - 1)$

Rewrite $9$ as $3^{2}$ .

$(x^{2} - 3^{2}) (x^{2} - 1)$

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 $b = 3$ .

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

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

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

Factor.

Tap for more steps…

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 $b = 1$ .

$(x + 3) (x - 3) ((x + 1) (x - 1))$

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x + 1) (x - 1)$

Replace the left side with the factored expression.

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

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

$x + 3 = 0$

$x - 3 = 0$

$x + 1 = 0$

$x - 1 = 0$

Set the first factor equal to $0$ and solve.

Tap for more steps…

Set the first factor equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set the next factor equal to $0$ and solve.

Tap for more steps…

Set the next factor equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set the next factor equal to $0$ and solve.

Tap for more steps…

Set the next factor equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set the next factor equal to $0$ and solve.

Tap for more steps…

Set the next factor equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$x = - 3, 3, - 1, 1$

Find the Roots (Zeros) x^4-10x^2+9=0

Need help with MATH HOMEWORK