Precalculus Examples

$(x^{2} + 10 x + 25) (x^{2} - 10 x + 25) - 49 = 0$

Step 1

Rewrite $(x^{2} + 10 x + 25) (x^{2} - 10 x + 25)$ as ${((x + 5) (x - 5))}^{2}$ .

${((x + 5) (x - 5))}^{2} - 49 = 0$

Step 2

Rewrite $49$ as $7^{2}$ .

${((x + 5) (x - 5))}^{2} - 7^{2} = 0$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (x + 5) (x - 5)$ and $b = 7$ .

$((x + 5) (x - 5) + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Expand $(x + 5) (x - 5)$ using the FOIL Method.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$(x (x - 5) + 5 (x - 5) + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.1.2

Apply the distributive property.

$(x \cdot x + x \cdot - 5 + 5 (x - 5) + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.1.3

Apply the distributive property.

$(x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5 + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.2

Combine the opposite terms in $x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5$ .

Tap for more steps...

Step 4.2.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

$(x \cdot x - 5 x + 5 x + 5 \cdot - 5 + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.2.2

Add $- 5 x$ and $5 x$ .

$(x \cdot x + 0 + 5 \cdot - 5 + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.2.3

Add $x \cdot x$ and $0$ .

$(x \cdot x + 5 \cdot - 5 + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.3

Simplify each term.

Tap for more steps...

Step 4.3.1

Multiply $x$ by $x$ .

$(x^{2} + 5 \cdot - 5 + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.3.2

Multiply $5$ by $- 5$ .

$(x^{2} - 25 + 7) ((x + 5) (x - 5) - 7) = 0$

Step 4.4

Add $- 25$ and $7$ .

$(x^{2} - 18) ((x + 5) (x - 5) - 7) = 0$

Step 4.5

Expand $(x + 5) (x - 5)$ using the FOIL Method.

Tap for more steps...

Step 4.5.1

Apply the distributive property.

$(x^{2} - 18) (x (x - 5) + 5 (x - 5) - 7) = 0$

Step 4.5.2

Apply the distributive property.

$(x^{2} - 18) (x \cdot x + x \cdot - 5 + 5 (x - 5) - 7) = 0$

Step 4.5.3

Apply the distributive property.

$(x^{2} - 18) (x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5 - 7) = 0$

Step 4.6

Combine the opposite terms in $x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5$ .

Tap for more steps...

Step 4.6.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

$(x^{2} - 18) (x \cdot x - 5 x + 5 x + 5 \cdot - 5 - 7) = 0$

Step 4.6.2

Add $- 5 x$ and $5 x$ .

$(x^{2} - 18) (x \cdot x + 0 + 5 \cdot - 5 - 7) = 0$

Step 4.6.3

Add $x \cdot x$ and $0$ .

$(x^{2} - 18) (x \cdot x + 5 \cdot - 5 - 7) = 0$

Step 4.7

Simplify each term.

Tap for more steps...

Step 4.7.1

Multiply $x$ by $x$ .

$(x^{2} - 18) (x^{2} + 5 \cdot - 5 - 7) = 0$

Step 4.7.2

Multiply $5$ by $- 5$ .

$(x^{2} - 18) (x^{2} - 25 - 7) = 0$

Step 4.8

Subtract $7$ from $- 25$ .

$(x^{2} - 18) (x^{2} - 32) = 0$

Step 5

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

$x^{2} - 18 = 0$

$x^{2} - 32 = 0$

Step 6

Set $x^{2} - 18$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.1

Set $x^{2} - 18$ equal to $0$ .

$x^{2} - 18 = 0$

Step 6.2

Solve $x^{2} - 18 = 0$ for $x$ .

Tap for more steps...

Step 6.2.1

Add $18$ to both sides of the equation.

$x^{2} = 18$

Step 6.2.2

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

$x = \pm \sqrt{18}$

Step 6.2.3

Simplify $\pm \sqrt{18}$ .

Tap for more steps...

Step 6.2.3.1

Rewrite $18$ as $3^{2} \cdot 2$ .

Tap for more steps...

Step 6.2.3.1.1

Factor $9$ out of $18$ .

$x = \pm \sqrt{9 (2)}$

Step 6.2.3.1.2

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2} \cdot 2}$

Step 6.2.3.2

Pull terms out from under the radical.

$x = \pm 3 \sqrt{2}$

Step 6.2.4

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

Tap for more steps...

Step 6.2.4.1

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

$x = 3 \sqrt{2}$

Step 6.2.4.2

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

$x = - 3 \sqrt{2}$

Step 6.2.4.3

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

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

Step 7

Set $x^{2} - 32$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $x^{2} - 32$ equal to $0$ .

$x^{2} - 32 = 0$

Step 7.2

Solve $x^{2} - 32 = 0$ for $x$ .

Tap for more steps...

Step 7.2.1

Add $32$ to both sides of the equation.

$x^{2} = 32$

Step 7.2.2

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

$x = \pm \sqrt{32}$

Step 7.2.3

Simplify $\pm \sqrt{32}$ .

Tap for more steps...

Step 7.2.3.1

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 7.2.3.1.1

Factor $16$ out of $32$ .

$x = \pm \sqrt{16 (2)}$

Step 7.2.3.1.2

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2} \cdot 2}$

Step 7.2.3.2

Pull terms out from under the radical.

$x = \pm 4 \sqrt{2}$

Step 7.2.4

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

Tap for more steps...

Step 7.2.4.1

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

$x = 4 \sqrt{2}$

Step 7.2.4.2

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

$x = - 4 \sqrt{2}$

Step 7.2.4.3

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

$x = 4 \sqrt{2}, - 4 \sqrt{2}$

Step 8

The final solution is all the values that make $(x^{2} - 18) (x^{2} - 32) = 0$ true.

$x = 3 \sqrt{2}, - 3 \sqrt{2}, 4 \sqrt{2}, - 4 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 3 \sqrt{2}, - 3 \sqrt{2}, 4 \sqrt{2}, - 4 \sqrt{2}$

Decimal Form:

$x = 4.24264068 \dots, - 4.24264068 \dots, 5.65685424 \dots, - 5.65685424 \dots$