Trigonometry Examples

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

Step 1

Simplify ${(x + 1)}^{2} (x - 2) + (x + 1) {(x - 2)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

${(x + 1)}^{2} x + {(x + 1)}^{2} \cdot - 2 + (x + 1) {(x - 2)}^{2} = 0$

Step 1.1.2

Move $- 2$ to the left of ${(x + 1)}^{2}$ .

${(x + 1)}^{2} x - 2 \cdot {(x + 1)}^{2} + (x + 1) {(x - 2)}^{2} = 0$

Step 1.1.3

Apply the distributive property.

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

Step 1.1.4

Multiply ${(x - 2)}^{2}$ by $1$ .

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

Step 1.2

Reorder factors in ${(x + 1)}^{2} x - 2 {(x + 1)}^{2} + x {(x - 2)}^{2} + {(x - 2)}^{2}$ .

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

Step 2

Factor the left side of the equation.

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Step 2.1

Factor ${(x + 1)}^{2}$ out of $x {(x + 1)}^{2} - 2 {(x + 1)}^{2}$ .

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Step 2.1.1

Factor ${(x + 1)}^{2}$ out of $x {(x + 1)}^{2}$ .

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

Step 2.1.2

Factor ${(x + 1)}^{2}$ out of $- 2 {(x + 1)}^{2}$ .

${(x + 1)}^{2} x + {(x + 1)}^{2} \cdot - 2 + x {(x - 2)}^{2} + {(x - 2)}^{2} = 0$

Step 2.1.3

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{2} x + {(x + 1)}^{2} \cdot - 2$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply by $1$ .

${(x + 1)}^{2} (x - 2) + {(x - 2)}^{2} x + {(x - 2)}^{2} \cdot 1 = 0$

Step 2.2.3

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} x + {(x - 2)}^{2} \cdot 1$ .

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

Step 2.3

Factor $(x + 1) (x - 2)$ out of ${(x + 1)}^{2} (x - 2) + {(x - 2)}^{2} (x + 1)$ .

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Step 2.3.1

Factor $(x + 1) (x - 2)$ out of ${(x + 1)}^{2} (x - 2)$ .

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

Step 2.3.2

Factor $(x + 1) (x - 2)$ out of ${(x - 2)}^{2} (x + 1)$ .

$(x + 1) (x - 2) (x + 1) + (x + 1) (x - 2) (x - 2) = 0$

Step 2.3.3

Factor $(x + 1) (x - 2)$ out of $(x + 1) (x - 2) (x + 1) + (x + 1) (x - 2) (x - 2)$ .

$(x + 1) (x - 2) (x + 1 + x - 2) = 0$

Step 2.4

Add $x$ and $x$ .

$(x + 1) (x - 2) (2 x + 1 - 2) = 0$

Step 2.5

Subtract $2$ from $1$ .

$(x + 1) (x - 2) (2 x - 1) = 0$

Step 3

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

$x + 1 = 0$

$x - 2 = 0$

$2 x - 1 = 0$

Step 4

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

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

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

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Step 5.1

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

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 6.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 6.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 7

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

$x = - 1, 2, \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 1, 2, \frac{1}{2}$

Decimal Form:

$x = - 1, 2, 0.5$