Find the Roots (Zeros) y=x^3-5x^2+9x-45

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Algebra

$y = x^{3} - 5 x^{2} + 9 x - 45$

Set $x^{3} - 5 x^{2} + 9 x - 45$ equal to $0$ .

$x^{3} - 5 x^{2} + 9 x - 45 = 0$

Solve for $x$ .

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Factor the left side of the equation.

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Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$(x^{3} - 5 x^{2}) + 9 x - 45 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $x - 5$ .

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

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

$x - 5 = 0$

$x^{2} + 9 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x^{2} + 9 = 0$

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

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

$x = \pm \sqrt{- 9}$

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

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Simplify the right side of the equation.

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Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 \cdot 9}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{9}$

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

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

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

$x = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$x = \pm 3 i$

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

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

$x = 3 i$

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

$x = - 3 i$

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

$x = 3 i, - 3 i$

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

$x = 5, 3 i, - 3 i$

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