Algebra Examples

${(x^{2} + 3)}^{2} + 21 = 10 x^{2} + 30$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $10 x^{2}$ from both sides of the equation.

${(x^{2} + 3)}^{2} + 21 - 10 x^{2} = 30$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(x^{2} + 3)}^{2}$ as $(x^{2} + 3) (x^{2} + 3)$ .

$(x^{2} + 3) (x^{2} + 3) + 21 - 10 x^{2} = 30$

Step 1.2.2

Expand $(x^{2} + 3) (x^{2} + 3)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$x^{2} (x^{2} + 3) + 3 (x^{2} + 3) + 21 - 10 x^{2} = 30$

Step 1.2.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 3 + 3 (x^{2} + 3) + 21 - 10 x^{2} = 30$

Step 1.2.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 3 + 3 x^{2} + 3 \cdot 3 + 21 - 10 x^{2} = 30$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 2} + x^{2} \cdot 3 + 3 x^{2} + 3 \cdot 3 + 21 - 10 x^{2} = 30$

Step 1.2.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 3 + 3 x^{2} + 3 \cdot 3 + 21 - 10 x^{2} = 30$

Step 1.2.3.1.2

Move $3$ to the left of $x^{2}$ .

$x^{4} + 3 \cdot x^{2} + 3 x^{2} + 3 \cdot 3 + 21 - 10 x^{2} = 30$

Step 1.2.3.1.3

Multiply $3$ by $3$ .

$x^{4} + 3 x^{2} + 3 x^{2} + 9 + 21 - 10 x^{2} = 30$

Step 1.2.3.2

Add $3 x^{2}$ and $3 x^{2}$ .

$x^{4} + 6 x^{2} + 9 + 21 - 10 x^{2} = 30$

Step 1.3

Subtract $10 x^{2}$ from $6 x^{2}$ .

$x^{4} - 4 x^{2} + 9 + 21 = 30$

Step 1.4

Add $9$ and $21$ .

$x^{4} - 4 x^{2} + 30 = 30$

Step 2

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

$u^{2} - 4 u + 30 = 30$

$u = x^{2}$

Step 3

Subtract $30$ from both sides of the equation.

$u^{2} - 4 u + 30 - 30 = 0$

Step 4

Combine the opposite terms in $u^{2} - 4 u + 30 - 30$ .

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Step 4.1

Subtract $30$ from $30$ .

$u^{2} - 4 u + 0 = 0$

Step 4.2

Add $u^{2} - 4 u$ and $0$ .

$u^{2} - 4 u = 0$

Step 5

Factor $u$ out of $u^{2} - 4 u$ .

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Step 5.1

Factor $u$ out of $u^{2}$ .

$u \cdot u - 4 u = 0$

Step 5.2

Factor $u$ out of $- 4 u$ .

$u \cdot u + u \cdot - 4 = 0$

Step 5.3

Factor $u$ out of $u \cdot u + u \cdot - 4$ .

$u (u - 4) = 0$

Step 6

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

$u = 0$

$u - 4 = 0$

Step 7

Set $u$ equal to $0$ .

$u = 0$

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

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

$u = 0, 4$

Step 10

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

$x^{2} = 0$

${(x^{2})}^{1} = 4$

Step 11

Solve the first equation for $x$ .

$x^{2} = 0$

Step 12

Solve the equation for $x$ .

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Step 12.1

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

$x = \pm \sqrt{0}$

Step 12.2

Simplify $\pm \sqrt{0}$ .

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Step 12.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 12.2.2

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

$x = \pm 0$

Step 12.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 13

Solve the second equation for $x$ .

${(x^{2})}^{1} = 4$

Step 14

Solve the equation for $x$ .

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Step 14.1

Remove parentheses.

$x^{2} = 4$

Step 14.2

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

$x = \pm \sqrt{4}$

Step 14.3

Simplify $\pm \sqrt{4}$ .

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Step 14.3.1

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

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

Step 14.3.2

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

$x = \pm 2$

Step 14.4

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

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Step 14.4.1

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

$x = 2$

Step 14.4.2

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

$x = - 2$

Step 14.4.3

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

$x = 2, - 2$

Step 15

The solution to $x^{4} - 4 x^{2} + 30 = 30$ is $x = 0, 2, - 2$ .

$x = 0, 2, - 2$