Finite Math Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as ${(x - 2)}^{3} - 6 {(x - 2)}^{2} + 9 (x - 2) = 0$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Multiply $9$ by $- 2$ .

${(x - 2)}^{3} - 6 {(x - 2)}^{2} + 9 x - 18 = 0$

Step 1.2.3

Factor the left side of the equation.

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Step 1.2.3.1

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{3} - 6 {(x - 2)}^{2}$ .

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Step 1.2.3.1.1

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

${(x - 2)}^{2} (x - 2) - 6 {(x - 2)}^{2} + 9 x - 18 = 0$

Step 1.2.3.1.2

Factor ${(x - 2)}^{2}$ out of $- 6 {(x - 2)}^{2}$ .

${(x - 2)}^{2} (x - 2) + {(x - 2)}^{2} \cdot - 6 + 9 x - 18 = 0$

Step 1.2.3.1.3

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} (x - 2) + {(x - 2)}^{2} \cdot - 6$ .

${(x - 2)}^{2} (x - 2 - 6) + 9 x - 18 = 0$

Step 1.2.3.2

Subtract $6$ from $- 2$ .

${(x - 2)}^{2} (x - 8) + 9 x - 18 = 0$

Step 1.2.3.3

Factor $9$ out of $9 x - 18$ .

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Step 1.2.3.3.1

Factor $9$ out of $9 x$ .

${(x - 2)}^{2} (x - 8) + 9 (x) - 18 = 0$

Step 1.2.3.3.2

Factor $9$ out of $- 18$ .

${(x - 2)}^{2} (x - 8) + 9 x + 9 \cdot - 2 = 0$

Step 1.2.3.3.3

Factor $9$ out of $9 x + 9 \cdot - 2$ .

${(x - 2)}^{2} (x - 8) + 9 (x - 2) = 0$

Step 1.2.3.4

Factor $x - 2$ out of ${(x - 2)}^{2} (x - 8) + 9 (x - 2)$ .

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Step 1.2.3.4.1

Factor $x - 2$ out of ${(x - 2)}^{2} (x - 8)$ .

$(x - 2) ((x - 2) (x - 8)) + 9 (x - 2) = 0$

Step 1.2.3.4.2

Factor $x - 2$ out of $9 (x - 2)$ .

$(x - 2) ((x - 2) (x - 8)) + (x - 2) \cdot 9 = 0$

Step 1.2.3.4.3

Factor $x - 2$ out of $(x - 2) ((x - 2) (x - 8)) + (x - 2) \cdot 9$ .

$(x - 2) ((x - 2) (x - 8) + 9) = 0$

Step 1.2.3.5

Expand $(x - 2) (x - 8)$ using the FOIL Method.

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Step 1.2.3.5.1

Apply the distributive property.

$(x - 2) (x (x - 8) - 2 (x - 8) + 9) = 0$

Step 1.2.3.5.2

Apply the distributive property.

$(x - 2) (x \cdot x + x \cdot - 8 - 2 (x - 8) + 9) = 0$

Step 1.2.3.5.3

Apply the distributive property.

$(x - 2) (x \cdot x + x \cdot - 8 - 2 x - 2 \cdot - 8 + 9) = 0$

Step 1.2.3.6

Simplify and combine like terms.

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Step 1.2.3.6.1

Simplify each term.

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Step 1.2.3.6.1.1

Multiply $x$ by $x$ .

$(x - 2) (x^{2} + x \cdot - 8 - 2 x - 2 \cdot - 8 + 9) = 0$

Step 1.2.3.6.1.2

Move $- 8$ to the left of $x$ .

$(x - 2) (x^{2} - 8 x - 2 x - 2 \cdot - 8 + 9) = 0$

Step 1.2.3.6.1.3

Multiply $- 2$ by $- 8$ .

$(x - 2) (x^{2} - 8 x - 2 x + 16 + 9) = 0$

Step 1.2.3.6.2

Subtract $2 x$ from $- 8 x$ .

$(x - 2) (x^{2} - 10 x + 16 + 9) = 0$

Step 1.2.3.7

Add $16$ and $9$ .

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

Step 1.2.3.8

Factor using the perfect square rule.

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Step 1.2.3.8.1

Rewrite $25$ as $5^{2}$ .

$(x - 2) (x^{2} - 10 x + 5^{2}) = 0$

Step 1.2.3.8.2

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

$10 x = 2 \cdot x \cdot 5$

Step 1.2.3.8.3

Rewrite the polynomial.

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

Step 1.2.3.8.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

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

Step 1.2.4

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

$x - 2 = 0$

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

Step 1.2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.6

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set ${(x - 5)}^{2}$ equal to $0$ .

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

Step 1.2.6.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 1.2.6.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.7

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

$x = 2, 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (5, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Remove parentheses.

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

Step 2.2.5

Simplify ${((0) - 2)}^{3} - 6 {((0) - 2)}^{2} + 9 ((0) - 2)$ .

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Step 2.2.5.1

Simplify each term.

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Step 2.2.5.1.1

Subtract $2$ from $0$ .

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

Step 2.2.5.1.2

Raise $- 2$ to the power of $3$ .

$y = - 8 - 6 {((0) - 2)}^{2} + 9 ((0) - 2)$

Step 2.2.5.1.3

Subtract $2$ from $0$ .

$y = - 8 - 6 {(- 2)}^{2} + 9 ((0) - 2)$

Step 2.2.5.1.4

Raise $- 2$ to the power of $2$ .

$y = - 8 - 6 \cdot 4 + 9 ((0) - 2)$

Step 2.2.5.1.5

Multiply $- 6$ by $4$ .

$y = - 8 - 24 + 9 ((0) - 2)$

Step 2.2.5.1.6

Subtract $2$ from $0$ .

$y = - 8 - 24 + 9 \cdot - 2$

Step 2.2.5.1.7

Multiply $9$ by $- 2$ .

$y = - 8 - 24 - 18$

Step 2.2.5.2

Simplify by subtracting numbers.

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Step 2.2.5.2.1

Subtract $24$ from $- 8$ .

$y = - 32 - 18$

Step 2.2.5.2.2

Subtract $18$ from $- 32$ .

$y = - 50$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 50)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (5, 0)$

y-intercept(s): $(0, - 50)$

Step 4