Basic Math Examples

$v^{4} + 10 = 11 v^{2}$

Step 1

Subtract $11 v^{2}$ from both sides of the equation.

$v^{4} + 10 - 11 v^{2} = 0$

Step 2

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

$u^{2} - 11 u + 10 = 0$

$u = v^{2}$

Step 3

Factor $u^{2} - 11 u + 10$ using the AC method.

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Step 3.1

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 $10$ and whose sum is $- 11$ .

$- 10, - 1$

Step 3.2

Write the factored form using these integers.

$(u - 10) (u - 1) = 0$

Step 4

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

$u - 10 = 0$

$u - 1 = 0$

Step 5

Set $u - 10$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u - 10$ equal to $0$ .

$u - 10 = 0$

Step 5.2

Add $10$ to both sides of the equation.

$u = 10$

Step 6

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

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Step 6.1

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

$u - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 7

The final solution is all the values that make $(u - 10) (u - 1) = 0$ true.

$u = 10, 1$

Step 8

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

$v^{2} = 10$

${(v^{2})}^{1} = 1$

Step 9

Solve the first equation for $v$ .

$v^{2} = 10$

Step 10

Solve the equation for $v$ .

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Step 10.1

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

$v = \pm \sqrt{10}$

Step 10.2

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

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Step 10.2.1

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

$v = \sqrt{10}$

Step 10.2.2

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

$v = - \sqrt{10}$

Step 10.2.3

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

$v = \sqrt{10}, - \sqrt{10}$

Step 11

Solve the second equation for $v$ .

${(v^{2})}^{1} = 1$

Step 12

Solve the equation for $v$ .

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Step 12.1

Remove parentheses.

$v^{2} = 1$

Step 12.2

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

$v = \pm \sqrt{1}$

Step 12.3

Any root of $1$ is $1$ .

$v = \pm 1$

Step 12.4

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

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Step 12.4.1

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

$v = 1$

Step 12.4.2

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

$v = - 1$

Step 12.4.3

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

$v = 1, - 1$

Step 13

The solution to $v^{4} - 11 v^{2} + 10 = 0$ is $v = \sqrt{10}, - \sqrt{10}, 1, - 1$ .

$v = \sqrt{10}, - \sqrt{10}, 1, - 1$

Step 14

The result can be shown in multiple forms.

Exact Form:

$v = \sqrt{10}, - \sqrt{10}, 1, - 1$

Decimal Form:

$v = 3.16227766 \dots, - 3.16227766 \dots, 1, - 1$