Algebra Examples

$3 x^{4} + 18 = 21 x^{2}$

Step 1

Subtract $21 x^{2}$ from both sides of the equation.

$3 x^{4} + 18 - 21 x^{2} = 0$

Step 2

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

$3 u^{2} - 21 u + 18 = 0$

$u = x^{2}$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $3$ out of $3 u^{2} - 21 u + 18$ .

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Step 3.1.1

Factor $3$ out of $3 u^{2}$ .

$3 (u^{2}) - 21 u + 18 = 0$

Step 3.1.2

Factor $3$ out of $- 21 u$ .

$3 (u^{2}) + 3 (- 7 u) + 18 = 0$

Step 3.1.3

Factor $3$ out of $18$ .

$3 u^{2} + 3 (- 7 u) + 3 \cdot 6 = 0$

Step 3.1.4

Factor $3$ out of $3 u^{2} + 3 (- 7 u)$ .

$3 (u^{2} - 7 u) + 3 \cdot 6 = 0$

Step 3.1.5

Factor $3$ out of $3 (u^{2} - 7 u) + 3 \cdot 6$ .

$3 (u^{2} - 7 u + 6) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $u^{2} - 7 u + 6$ using the AC method.

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Step 3.2.1.1

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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 3.2.1.2

Write the factored form using these integers.

$3 ((u - 6) (u - 1)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$3 (u - 6) (u - 1) = 0$

Step 4

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

$u - 6 = 0$

$u - 1 = 0$

Step 5

Set $u - 6$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u - 6$ equal to $0$ .

$u - 6 = 0$

Step 5.2

Add $6$ to both sides of the equation.

$u = 6$

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 $3 (u - 6) (u - 1) = 0$ true.

$u = 6, 1$

Step 8

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

$x^{2} = 6$

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

Step 9

Solve the first equation for $x$ .

$x^{2} = 6$

Step 10

Solve the equation for $x$ .

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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.

$x = \pm \sqrt{6}$

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.

$x = \sqrt{6}$

Step 10.2.2

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

$x = - \sqrt{6}$

Step 10.2.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 11

Solve the second equation for $x$ .

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

Step 12

Solve the equation for $x$ .

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Step 12.1

Remove parentheses.

$x^{2} = 1$

Step 12.2

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

$x = \pm \sqrt{1}$

Step 12.3

Any root of $1$ is $1$ .

$x = \pm 1$

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.

$x = 1$

Step 12.4.2

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

$x = - 1$

Step 12.4.3

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

$x = 1, - 1$

Step 13

The solution to $3 x^{4} - 21 x^{2} + 18 = 0$ is $x = \sqrt{6}, - \sqrt{6}, 1, - 1$ .

$x = \sqrt{6}, - \sqrt{6}, 1, - 1$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{6}, - \sqrt{6}, 1, - 1$

Decimal Form:

$x = 2.44948974 \dots, - 2.44948974 \dots, 1, - 1$