Algebra Examples

$y = (x - 1) (x^{2} - 5 x + 6)$

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 $x$ .

$0 = (x - 1) (x^{2} - 5 x + 6)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x - 1) (x^{2} - 5 x + 6) = 0$ .

$(x - 1) (x^{2} - 5 x + 6) = 0$

Step 1.2.2

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

$x - 1 = 0$

$x^{2} - 5 x + 6 = 0$

Step 1.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.4

Set $x^{2} - 5 x + 6$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 5 x + 6$ equal to $0$ .

$x^{2} - 5 x + 6 = 0$

Step 1.2.4.2

Solve $x^{2} - 5 x + 6 = 0$ for $x$ .

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Step 1.2.4.2.1

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 1.2.4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2.4.2.1.2

Write the factored form using these integers.

$(x - 3) (x - 2) = 0$

Step 1.2.4.2.2

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

$x - 3 = 0$

$x - 2 = 0$

Step 1.2.4.2.3

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

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Step 1.2.4.2.3.1

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

$x - 3 = 0$

Step 1.2.4.2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.4.2.4

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

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Step 1.2.4.2.4.1

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

$x - 2 = 0$

Step 1.2.4.2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.4.2.5

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

$x = 3, 2$

Step 1.2.5

The final solution is all the values that make $(x - 1) (x^{2} - 5 x + 6) = 0$ true.

$x = 1, 3, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (3, 0), (2, 0)$

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 $y$ .

$y = ((0) - 1) ({(0)}^{2} - 5 \cdot 0 + 6)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0 - 1) ({(0)}^{2} - 5 \cdot 0 + 6)$

Step 2.2.2

Remove parentheses.

$y = (0 - 1) (0^{2} - 5 \cdot 0 + 6)$

Step 2.2.3

Remove parentheses.

$y = ((0) - 1) ({(0)}^{2} - 5 \cdot 0 + 6)$

Step 2.2.4

Simplify $((0) - 1) ({(0)}^{2} - 5 \cdot 0 + 6)$ .

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Step 2.2.4.1

Subtract $1$ from $0$ .

$y = - 1 ({(0)}^{2} - 5 \cdot 0 + 6)$

Step 2.2.4.2

Simplify each term.

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Step 2.2.4.2.1

Raising $0$ to any positive power yields $0$ .

$y = - 1 (0 - 5 \cdot 0 + 6)$

Step 2.2.4.2.2

Multiply $- 5$ by $0$ .

$y = - 1 (0 + 0 + 6)$

Step 2.2.4.3

Simplify the expression.

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Step 2.2.4.3.1

Add $0$ and $0$ .

$y = - 1 (0 + 6)$

Step 2.2.4.3.2

Add $0$ and $6$ .

$y = - 1 \cdot 6$

Step 2.2.4.3.3

Multiply $- 1$ by $6$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(1, 0), (3, 0), (2, 0)$

y-intercept(s): $(0, - 6)$

Step 4