Precalculus Examples

$(3 x + 5) {(x^{2} - 6 x + 9)}^{2}$

Step 1

Set $(3 x + 5) {(x^{2} - 6 x + 9)}^{2}$ equal to $0$ .

$(3 x + 5) {(x^{2} - 6 x + 9)}^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$3 x + 5 = 0$

${(x^{2} - 6 x + 9)}^{2} = 0$

Step 2.2

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 2.2.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 2.2.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 2.2.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 2.2.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 2.2.2.2.2

Simplify the left side.

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Step 2.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 2.2.2.2.3

Simplify the right side.

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Step 2.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 2.3

Set ${(x^{2} - 6 x + 9)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set ${(x^{2} - 6 x + 9)}^{2}$ equal to $0$ .

${(x^{2} - 6 x + 9)}^{2} = 0$

Step 2.3.2

Solve ${(x^{2} - 6 x + 9)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $x^{2} - 6 x + 9$ equal to $0$ .

$x^{2} - 6 x + 9 = 0$

Step 2.3.2.2

Solve for $x$ .

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Step 2.3.2.2.1

Factor using the perfect square rule.

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Step 2.3.2.2.1.1

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

$x^{2} - 6 x + 3^{2} = 0$

Step 2.3.2.2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 2.3.2.2.1.3

Rewrite the polynomial.

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

Step 2.3.2.2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 2.3.2.2.2

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

$x - 3 = 0$

Step 2.3.2.2.3

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

The final solution is all the values that make $(3 x + 5) {(x^{2} - 6 x + 9)}^{2} = 0$ true.

$x = - \frac{5}{3}, 3$

Step 3