Algebra Examples

$y = 4 x^{4} - 11 x^{2} - 3$

Step 1

Set $4 x^{4} - 11 x^{2} - 3$ equal to $0$ .

$4 x^{4} - 11 x^{2} - 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$4 u^{2} - 11 u - 3 = 0$

$u = x^{2}$

Step 2.2

Factor by grouping.

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Step 2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = - 11$ .

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Step 2.2.1.1

Factor $- 11$ out of $- 11 u$ .

$4 u^{2} - 11 u - 3 = 0$

Step 2.2.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$4 u^{2} + (1 - 12) u - 3 = 0$

Step 2.2.1.3

Apply the distributive property.

$4 u^{2} + 1 u - 12 u - 3 = 0$

Step 2.2.1.4

Multiply $u$ by $1$ .

$4 u^{2} + u - 12 u - 3 = 0$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$(4 u^{2} + u) - 12 u - 3 = 0$

Step 2.2.2.2

Factor out the greatest common factor (GCF) from each group.

$u (4 u + 1) - 3 (4 u + 1) = 0$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $4 u + 1$ .

$(4 u + 1) (u - 3) = 0$

Step 2.3

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

$4 u + 1 = 0$

$u - 3 = 0$

Step 2.4

Set $4 u + 1$ equal to $0$ and solve for $u$ .

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Step 2.4.1

Set $4 u + 1$ equal to $0$ .

$4 u + 1 = 0$

Step 2.4.2

Solve $4 u + 1 = 0$ for $u$ .

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Step 2.4.2.1

Subtract $1$ from both sides of the equation.

$4 u = - 1$

Step 2.4.2.2

Divide each term in $4 u = - 1$ by $4$ and simplify.

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Step 2.4.2.2.1

Divide each term in $4 u = - 1$ by $4$ .

$\frac{4 u}{4} = \frac{- 1}{4}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{- 1}{4}$

Step 2.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{4}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{4}$

Step 2.5

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

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Step 2.5.1

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

$u - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$u = 3$

Step 2.6

The final solution is all the values that make $(4 u + 1) (u - 3) = 0$ true.

$u = - \frac{1}{4}, 3$

Step 2.7

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

$x^{2} = - \frac{1}{4}$

${(x^{2})}^{1} = 3$

Step 2.8

Solve the first equation for $x$ .

$x^{2} = - \frac{1}{4}$

Step 2.9

Solve the equation for $x$ .

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Step 2.9.1

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

$x = \pm \sqrt{- \frac{1}{4}}$

Step 2.9.2

Simplify $\pm \sqrt{- \frac{1}{4}}$ .

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Step 2.9.2.1

Rewrite $- \frac{1}{4}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2}}$

Step 2.9.2.2

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

$x = \pm \frac{i}{2}$

Step 2.9.3

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

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Step 2.9.3.1

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

$x = \frac{i}{2}$

Step 2.9.3.2

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

$x = - \frac{i}{2}$

Step 2.9.3.3

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

$x = \frac{i}{2}, - \frac{i}{2}$

Step 2.10

Solve the second equation for $x$ .

${(x^{2})}^{1} = 3$

Step 2.11

Solve the equation for $x$ .

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Step 2.11.1

Remove parentheses.

$x^{2} = 3$

Step 2.11.2

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

$x = \pm \sqrt{3}$

Step 2.11.3

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

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Step 2.11.3.1

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

$x = \sqrt{3}$

Step 2.11.3.2

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

$x = - \sqrt{3}$

Step 2.11.3.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 2.12

The solution to $4 x^{4} - 11 x^{2} - 3 = 0$ is $x = \frac{i}{2}, - \frac{i}{2}, \sqrt{3}, - \sqrt{3}$ .

$x = \frac{i}{2}, - \frac{i}{2}, \sqrt{3}, - \sqrt{3}$

Step 3