Precalculus Examples

$f (x) = 50 x^{4} - 150 x^{3} - 450 x^{2} + 1350 x$

Step 1

Set $50 x^{4} - 150 x^{3} - 450 x^{2} + 1350 x$ equal to $0$ .

$50 x^{4} - 150 x^{3} - 450 x^{2} + 1350 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $50 x$ out of $50 x^{4} - 150 x^{3} - 450 x^{2} + 1350 x$ .

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Step 2.1.1.1

Factor $50 x$ out of $50 x^{4}$ .

$50 x (x^{3}) - 150 x^{3} - 450 x^{2} + 1350 x = 0$

Step 2.1.1.2

Factor $50 x$ out of $- 150 x^{3}$ .

$50 x (x^{3}) + 50 x (- 3 x^{2}) - 450 x^{2} + 1350 x = 0$

Step 2.1.1.3

Factor $50 x$ out of $- 450 x^{2}$ .

$50 x (x^{3}) + 50 x (- 3 x^{2}) + 50 x (- 9 x) + 1350 x = 0$

Step 2.1.1.4

Factor $50 x$ out of $1350 x$ .

$50 x (x^{3}) + 50 x (- 3 x^{2}) + 50 x (- 9 x) + 50 x (27) = 0$

Step 2.1.1.5

Factor $50 x$ out of $50 x (x^{3}) + 50 x (- 3 x^{2})$ .

$50 x (x^{3} - 3 x^{2}) + 50 x (- 9 x) + 50 x (27) = 0$

Step 2.1.1.6

Factor $50 x$ out of $50 x (x^{3} - 3 x^{2}) + 50 x (- 9 x)$ .

$50 x (x^{3} - 3 x^{2} - 9 x) + 50 x (27) = 0$

Step 2.1.1.7

Factor $50 x$ out of $50 x (x^{3} - 3 x^{2} - 9 x) + 50 x (27)$ .

$50 x (x^{3} - 3 x^{2} - 9 x + 27) = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$50 x ((x^{3} - 3 x^{2}) - 9 x + 27) = 0$

Step 2.1.2.2

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

$50 x (x^{2} (x - 3) - 9 (x - 3)) = 0$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$50 x ((x - 3) (x^{2} - 9)) = 0$

Step 2.1.4

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

$50 x ((x - 3) (x^{2} - 3^{2})) = 0$

Step 2.1.5

Factor.

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Step 2.1.5.1

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 $b = 3$ .

$50 x ((x - 3) ((x + 3) (x - 3))) = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

$50 x ((x - 3) (x + 3) (x - 3)) = 0$

Step 2.1.6

Factor.

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Step 2.1.6.1

Combine exponents.

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Step 2.1.6.1.1

Raise $x - 3$ to the power of $1$ .

$50 x ((x - 3) (x - 3) (x + 3)) = 0$

Step 2.1.6.1.2

Raise $x - 3$ to the power of $1$ .

$50 x ((x - 3) (x - 3) (x + 3)) = 0$

Step 2.1.6.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$50 x ({(x - 3)}^{1 + 1} (x + 3)) = 0$

Step 2.1.6.1.4

Add $1$ and $1$ .

$50 x ({(x - 3)}^{2} (x + 3)) = 0$

Step 2.1.6.2

Remove unnecessary parentheses.

$50 x {(x - 3)}^{2} (x + 3) = 0$

Step 2.2

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

$x = 0$

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

$x + 3 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

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

$x - 3 = 0$

Step 2.4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.5

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

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Step 2.5.1

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

$x + 3 = 0$

Step 2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

The final solution is all the values that make $50 x {(x - 3)}^{2} (x + 3) = 0$ true.

$x = 0, 3, - 3$

Step 3