Solve for t t^4-21t^2+80=0

$t^{4} - 21 t^{2} + 80 = 0$

Step 1

Substitute $u = t^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 21 u + 80 = 0$

$u = t^{2}$

Step 2

Factor $u^{2} - 21 u + 80$ 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 $80$ and whose sum is $- 21$ .

$- 16, - 5$

Write the factored form using these integers.

$(u - 16) (u - 5) = 0$

Step 3

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

$u - 16 = 0$

$u - 5 = 0$

Step 4

Set $u - 16$ equal to $0$ and solve for $u$ .

Set $u - 16$ equal to $0$ .

$u - 16 = 0$

Add $16$ to both sides of the equation.

$u = 16$

Step 5

Set $u - 5$ equal to $0$ and solve for $u$ .

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Add $5$ to both sides of the equation.

$u = 5$

Step 6

The final solution is all the values that make $(u - 16) (u - 5) = 0$ true.

$u = 16, 5$

Step 7

Substitute the real value of $u = t^{2}$ back into the solved equation.

$t^{2} = 16$

${(t^{2})}^{1} = 5$

Step 8

Solve the first equation for $t$ .

$t^{2} = 16$

Step 9

Solve the equation for $t$ .

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

$t = \pm \sqrt{16}$

Simplify $\pm \sqrt{16}$ .

Rewrite $16$ as $4^{2}$ .

$t = \pm \sqrt{4^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm 4$

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.

$t = 4$

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

$t = - 4$

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

$t = 4, - 4$

Step 10

Solve the second equation for $t$ .

${(t^{2})}^{1} = 5$

Step 11

Solve the equation for $t$ .

Remove parentheses.

$t^{2} = 5$

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

$t = \pm \sqrt{5}$

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.

$t = \sqrt{5}$

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

$t = - \sqrt{5}$

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

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

Step 12

The solution to $t^{4} - 21 t^{2} + 80 = 0$ is $t = 4, - 4, \sqrt{5}, - \sqrt{5}$ .

$t = 4, - 4, \sqrt{5}, - \sqrt{5}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$t = 4, - 4, \sqrt{5}, - \sqrt{5}$

Decimal Form:

$t = 4, - 4, 2.23606797 \dots, - 2.23606797 \dots$

Solve for t t^4-21t^2+80=0

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