Algebra Examples

$f (x) = 2 x^{4} + x^{3} - 7 x^{2} - 3 x + 3$

Step 1

Set $2 x^{4} + x^{3} - 7 x^{2} - 3 x + 3$ equal to $0$ .

$2 x^{4} + x^{3} - 7 x^{2} - 3 x + 3 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Factor the left side of the equation.

Tap for more steps...

Step 2.1.1

Regroup terms.

$x^{3} - 3 x + 2 x^{4} - 7 x^{2} + 3 = 0$

Step 2.1.2

Factor $x$ out of $x^{3} - 3 x$ .

Tap for more steps...

Step 2.1.2.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 3 x + 2 x^{4} - 7 x^{2} + 3 = 0$

Step 2.1.2.2

Factor $x$ out of $- 3 x$ .

$x \cdot x^{2} + x \cdot - 3 + 2 x^{4} - 7 x^{2} + 3 = 0$

Step 2.1.2.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 3$ .

$x (x^{2} - 3) + 2 x^{4} - 7 x^{2} + 3 = 0$

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Factor by grouping.

Tap for more steps...

Step 2.1.5.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

Tap for more steps...

Step 2.1.5.1.1

Factor $- 7$ out of $- 7 u$ .

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

Step 2.1.5.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$x (x^{2} - 3) + 2 u^{2} + (- 1 - 6) u + 3 = 0$

Step 2.1.5.1.3

Apply the distributive property.

$x (x^{2} - 3) + 2 u^{2} - 1 u - 6 u + 3 = 0$

Step 2.1.5.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.5.2.1

Group the first two terms and the last two terms.

$x (x^{2} - 3) + 2 u^{2} - 1 u - 6 u + 3 = 0$

Step 2.1.5.2.2

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

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

Step 2.1.5.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

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

Step 2.1.6

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

$x (x^{2} - 3) + (2 x^{2} - 1) (x^{2} - 3) = 0$

Step 2.1.7

Factor $x^{2} - 3$ out of $x (x^{2} - 3) + (2 x^{2} - 1) (x^{2} - 3)$ .

Tap for more steps...

Step 2.1.7.1

Factor $x^{2} - 3$ out of $x (x^{2} - 3)$ .

$(x^{2} - 3) x + (2 x^{2} - 1) (x^{2} - 3) = 0$

Step 2.1.7.2

Factor $x^{2} - 3$ out of $(2 x^{2} - 1) (x^{2} - 3)$ .

$(x^{2} - 3) x + (x^{2} - 3) (2 x^{2} - 1) = 0$

Step 2.1.7.3

Factor $x^{2} - 3$ out of $(x^{2} - 3) x + (x^{2} - 3) (2 x^{2} - 1)$ .

$(x^{2} - 3) (x + 2 x^{2} - 1) = 0$

Step 2.1.8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 2.1.9

Factor by grouping.

Tap for more steps...

Step 2.1.9.1

Reorder terms.

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

Step 2.1.9.2

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

Tap for more steps...

Step 2.1.9.2.1

Multiply by $1$ .

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

Step 2.1.9.2.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 2.1.9.2.3

Apply the distributive property.

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

Step 2.1.9.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.9.3.1

Group the first two terms and the last two terms.

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

Step 2.1.9.3.2

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

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

Step 2.1.9.4

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

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

Step 2.1.10

Factor.

Tap for more steps...

Step 2.1.10.1

Replace all occurrences of $u$ with $x$ .

$(x^{2} - 3) ((2 x - 1) (x + 1)) = 0$

Step 2.1.10.2

Remove unnecessary parentheses.

$(x^{2} - 3) (2 x - 1) (x + 1) = 0$

Step 2.2

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

$x^{2} - 3 = 0$

$2 x - 1 = 0$

$x + 1 = 0$

Step 2.3

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.1

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

$x^{2} - 3 = 0$

Step 2.3.2

Solve $x^{2} - 3 = 0$ for $x$ .

Tap for more steps...

Step 2.3.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 2.3.2.2

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

$x = \pm \sqrt{3}$

Step 2.3.2.3

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

Tap for more steps...

Step 2.3.2.3.1

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

$x = \sqrt{3}$

Step 2.3.2.3.2

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

$x = - \sqrt{3}$

Step 2.3.2.3.3

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

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$2 x - 1 = 0$

Step 2.4.2

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

Tap for more steps...

Step 2.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.4.2.2

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

Tap for more steps...

Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$x + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.6

The final solution is all the values that make $(x^{2} - 3) (2 x - 1) (x + 1) = 0$ true.

$x = \sqrt{3}, - \sqrt{3}, \frac{1}{2}, - 1$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - \sqrt{3}, \frac{1}{2}, - 1$

Decimal Form:

$x = 1.73205080 \dots, - 1.73205080 \dots, 0.5, - 1$

Step 4