Basic Math Examples

$11 = k^{2} - 1 + k^{4} - 2 k^{2} + 1 - 1$

Step 1

Rewrite the equation as $k^{2} - 1 + k^{4} - 2 k^{2} + 1 - 1 = 11$ .

$k^{2} - 1 + k^{4} - 2 k^{2} + 1 - 1 = 11$

Step 2

Simplify $k^{2} - 1 + k^{4} - 2 k^{2} + 1 - 1$ .

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Step 2.1

Combine the opposite terms in $k^{2} - 1 + k^{4} - 2 k^{2} + 1 - 1$ .

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Step 2.1.1

Add $- 1$ and $1$ .

$k^{2} + k^{4} - 2 k^{2} + 0 - 1 = 11$

Step 2.1.2

Add $k^{2} + k^{4} - 2 k^{2}$ and $0$ .

$k^{2} + k^{4} - 2 k^{2} - 1 = 11$

Step 2.2

Subtract $2 k^{2}$ from $k^{2}$ .

$k^{4} - k^{2} - 1 = 11$

Step 3

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

$u^{2} - u - 1 = 11$

$u = k^{2}$

Step 4

Subtract $11$ from both sides of the equation.

$u^{2} - u - 1 - 11 = 0$

Step 5

Subtract $11$ from $- 1$ .

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

Step 6

Factor $u^{2} - u - 12$ using the AC method.

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Step 6.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 6.2

Write the factored form using these integers.

$(u - 4) (u + 3) = 0$

Step 7

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

$u - 4 = 0$

$u + 3 = 0$

Step 8

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$u = 4$

Step 9

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

Step 9.2

Subtract $3$ from both sides of the equation.

$u = - 3$

Step 10

The final solution is all the values that make $(u - 4) (u + 3) = 0$ true.

$u = 4, - 3$

Step 11

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

$k^{2} = 4$

${(k^{2})}^{1} = - 3$

Step 12

Solve the first equation for $k$ .

$k^{2} = 4$

Step 13

Solve the equation for $k$ .

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Step 13.1

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

$k = \pm \sqrt{4}$

Step 13.2

Simplify $\pm \sqrt{4}$ .

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Step 13.2.1

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

$k = \pm \sqrt{2^{2}}$

Step 13.2.2

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

$k = \pm 2$

Step 13.3

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

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Step 13.3.1

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

$k = 2$

Step 13.3.2

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

$k = - 2$

Step 13.3.3

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

$k = 2, - 2$

Step 14

Solve the second equation for $k$ .

${(k^{2})}^{1} = - 3$

Step 15

Solve the equation for $k$ .

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Step 15.1

Remove parentheses.

$k^{2} = - 3$

Step 15.2

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

$k = \pm \sqrt{- 3}$

Step 15.3

Simplify $\pm \sqrt{- 3}$ .

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Step 15.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$k = \pm \sqrt{- 1 (3)}$

Step 15.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$k = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 15.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$k = \pm i \sqrt{3}$

Step 15.4

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

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Step 15.4.1

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

$k = i \sqrt{3}$

Step 15.4.2

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

$k = - i \sqrt{3}$

Step 15.4.3

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

$k = i \sqrt{3}, - i \sqrt{3}$

Step 16

The solution to $k^{4} - k^{2} - 1 = 11$ is $k = 2, - 2, i \sqrt{3}, - i \sqrt{3}$ .

$k = 2, - 2, i \sqrt{3}, - i \sqrt{3}$