Solve by Factoring a^4-12a^2+35=0

$a^{4} - 12 a^{2} + 35 = 0$

Step 1

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

${(a^{2})}^{2} - 12 a^{2} + 35 = 0$

Step 2

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

$u^{2} - 12 u + 35 = 0$

Step 3

Factor $u^{2} - 12 u + 35$ 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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Write the factored form using these integers.

$(u - 7) (u - 5) = 0$

Step 4

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

$(a^{2} - 7) (a^{2} - 5) = 0$

Step 5

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

$a^{2} - 7 = 0$

$a^{2} - 5 = 0$

Step 6

Set $a^{2} - 7$ equal to $0$ and solve for $a$ .

Set $a^{2} - 7$ equal to $0$ .

$a^{2} - 7 = 0$

Solve $a^{2} - 7 = 0$ for $a$ .

Add $7$ to both sides of the equation.

$a^{2} = 7$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{7}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$a = \sqrt{7}$

Next, use the negative value of the $\pm$ to find the second solution.

$a = - \sqrt{7}$

The complete solution is the result of both the positive and negative portions of the solution.

$a = \sqrt{7}, - \sqrt{7}$

Step 7

Set $a^{2} - 5$ equal to $0$ and solve for $a$ .

Set $a^{2} - 5$ equal to $0$ .

$a^{2} - 5 = 0$

Solve $a^{2} - 5 = 0$ for $a$ .

Add $5$ to both sides of the equation.

$a^{2} = 5$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$a = \sqrt{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$a = - \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$a = \sqrt{5}, - \sqrt{5}$

Step 8

The final solution is all the values that make $(a^{2} - 7) (a^{2} - 5) = 0$ true.

$a = \sqrt{7}, - \sqrt{7}, \sqrt{5}, - \sqrt{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$a = \sqrt{7}, - \sqrt{7}, \sqrt{5}, - \sqrt{5}$

Decimal Form:

$a = 2.64575131 \dots, - 2.64575131 \dots, 2.23606797 \dots, - 2.23606797 \dots$

Solve by Factoring a^4-12a^2+35=0

Related Questions

Support LCMGCF.com by disabling your adblocker.