Algebra Examples

$4 x^{4} + 15 x^{2} - 4 = 0$

Step 1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$4 {(x^{2})}^{2} + 15 x^{2} - 4 = 0$

Step 2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$4 u^{2} + 15 u - 4 = 0$

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

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 $a \cdot c = 4 \cdot - 4 = - 16$ and whose sum is $b = 15$ .

Tap for more steps...

Step 3.1.1

Factor $15$ out of $15 u$ .

$4 u^{2} + 15 (u) - 4 = 0$

Step 3.1.2

Rewrite $15$ as $- 1$ plus $16$

$4 u^{2} + (- 1 + 16) u - 4 = 0$

Step 3.1.3

Apply the distributive property.

$4 u^{2} - 1 u + 16 u - 4 = 0$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 1 u) + 16 u - 4 = 0$

Step 3.2.2

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

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

Step 3.3

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

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

Step 4

Replace all occurrences of $u$ with $x^{2}$ .

$(4 x^{2} - 1) (x^{2} + 4) = 0$

Step 5

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$({(2 x)}^{2} - 1) (x^{2} + 4) = 0$

Step 6

Rewrite $1$ as $1^{2}$ .

$({(2 x)}^{2} - 1^{2}) (x^{2} + 4) = 0$

Step 7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$(2 x + 1) (2 x - 1) (x^{2} + 4) = 0$

Step 8

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

$2 x + 1 = 0$

$2 x - 1 = 0$

$x^{2} + 4 = 0$

Step 9

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 9.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 9.2

Solve $2 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 9.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 9.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

Tap for more steps...

Step 9.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 9.2.2.2

Simplify the left side.

Tap for more steps...

Step 9.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 9.2.2.3

Simplify the right side.

Tap for more steps...

Step 9.2.2.3.1

Move the negative in front of the fraction.

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

Step 10

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 10.2

Solve $2 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 10.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 10.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

Tap for more steps...

Step 10.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 10.2.2.2

Simplify the left side.

Tap for more steps...

Step 10.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 11

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 11.1

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 11.2

Solve $x^{2} + 4 = 0$ for $x$ .

Tap for more steps...

Step 11.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 11.2.2

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

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

Step 11.2.3

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

Tap for more steps...

Step 11.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 11.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 11.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 11.2.3.4

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

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

Step 11.2.3.5

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

$x = \pm i \cdot 2$

Step 11.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 11.2.4

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

Tap for more steps...

Step 11.2.4.1

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

$x = 2 i$

Step 11.2.4.2

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

$x = - 2 i$

Step 11.2.4.3

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

$x = 2 i, - 2 i$

Step 12

The final solution is all the values that make $(2 x + 1) (2 x - 1) (x^{2} + 4) = 0$ true.

$x = - \frac{1}{2}, \frac{1}{2}, 2 i, - 2 i$