Calculus Examples

${(x^{2} - 9)}^{2} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 1.3

Apply the product rule to $(x + 3) (x - 3)$ .

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

Step 2

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

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

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

Step 3

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 3.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 3.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

$x = - 3, 3$