Algebra Examples

2 x^{4} - 9 x^{2} + 4 = 0

$2 x^{4} - 9 x^{2} + 4 = 0$

Step 1

Substitute

u = x^{2}

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

2 u^{2} - 9 u + 4 = 0

$2 u^{2} - 9 u + 4 = 0$

u = x^{2}

$u = x^{2}$

Step 2

Factor by grouping.

Tap for more steps...

Step 2.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 2 \cdot 4 = 8

$a \cdot c = 2 \cdot 4 = 8$ and whose sum is

b = - 9

$b = - 9$ .

Tap for more steps...

Step 2.1.1

Factor

- 9

$- 9$ out of

- 9 u

$- 9 u$ .

2 u^{2} - 9 u + 4 = 0

$2 u^{2} - 9 u + 4 = 0$

Step 2.1.2

Rewrite

- 9

$- 9$ as

- 1

$- 1$ plus

- 8

$- 8$

2 u^{2} + (- 1 - 8) u + 4 = 0

$2 u^{2} + (- 1 - 8) u + 4 = 0$

Step 2.1.3

Apply the distributive property.

2 u^{2} - 1 u - 8 u + 4 = 0

$2 u^{2} - 1 u - 8 u + 4 = 0$

2 u^{2} - 1 u - 8 u + 4 = 0

$2 u^{2} - 1 u - 8 u + 4 = 0$

Step 2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.1

Group the first two terms and the last two terms.

(2 u^{2} - 1 u) - 8 u + 4 = 0

$(2 u^{2} - 1 u) - 8 u + 4 = 0$

Step 2.2.2

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

$u (2 u - 1) - 4 (2 u - 1) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor,

$2 u - 1$ .

$(2 u - 1) (u - 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$2 u - 1 = 0$

$u - 4 = 0$

Step 4

Set

$2 u - 1$ equal to

$0$ and solve for

$u$ .

Tap for more steps...

Step 4.1

Set

$2 u - 1$ equal to

$0$ .

$2 u - 1 = 0$

Step 4.2

Solve

$2 u - 1 = 0$ for

$u$ .

Tap for more steps...

Step 4.2.1

Add

$1$ to both sides of the equation.

$2 u = 1$

Step 4.2.2

Divide each term in

$2 u = 1$ by

$2$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in

$2 u = 1$ by

$2$ .

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

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of

$2$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide

$u$ by

$1$ .

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

Step 5

Set

$u - 4$ equal to

$0$ and solve for

$u$ .

Tap for more steps...

Step 5.1

Set

$u - 4$ equal to

$0$ .

$u - 4 = 0$

Step 5.2

Add

$4$ to both sides of the equation.

$u = 4$

Step 6

The final solution is all the values that make

$(2 u - 1) (u - 4) = 0$ true.

$u = \frac{1}{2}, 4$

Step 7

Substitute the real value of

$u = x^{2}$ back into the solved equation.

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

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

Step 8

Solve the first equation for

$x$ .

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

Step 9

Solve the equation for

$x$ .

Tap for more steps...

Step 9.1

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

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

Step 9.2

Simplify

$\pm \sqrt{\frac{1}{2}}$ .

Tap for more steps...

Step 9.2.1

Rewrite

$\sqrt{\frac{1}{2}}$ as

$\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 9.2.2

Any root of

$1$ is

$1$ .

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

Step 9.2.3

Multiply

$\frac{1}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9.2.4

Combine and simplify the denominator.

Tap for more steps...

Step 9.2.4.1

Multiply

$\frac{1}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 9.2.4.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 9.2.4.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 9.2.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.2.4.5

Add

$1$ and

$1$ .

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

Step 9.2.4.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

Tap for more steps...

Step 9.2.4.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 9.2.4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 9.2.4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 9.2.4.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 9.2.4.6.4.1

Cancel the common factor.

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

Step 9.2.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Step 9.2.4.6.5

Evaluate the exponent.

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

Step 9.3

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

Tap for more steps...

Step 9.3.1

First, use the positive value of the

$\pm$ to find the first solution.

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

Step 9.3.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 9.3.3

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

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

Step 10

Solve the second equation for

$x$ .

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

Step 11

Solve the equation for

$x$ .

Tap for more steps...

Step 11.1

Remove parentheses.

$x^{2} = 4$

Step 11.2

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

$x = \pm \sqrt{4}$

Step 11.3

Simplify

$\pm \sqrt{4}$ .

Tap for more steps...

Step 11.3.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 11.3.2

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

$x = \pm 2$

Step 11.4

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

Tap for more steps...

Step 11.4.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = 2$

Step 11.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - 2$

Step 11.4.3

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

$x = 2, - 2$

Step 12

The solution to

$2 x^{4} - 9 x^{2} + 4 = 0$ is

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}, 2, - 2$ .

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

Step 13

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.70710678 \dots, - 0.70710678 \dots, 2, - 2$

Step 14