Algebra Examples

${(x - 3)}^{2} + {(x + 1)}^{2} = {(x - 2)}^{2} + {(x + 3)}^{2}$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract ${(x - 2)}^{2}$ from both sides of the equation.

${(x - 3)}^{2} + {(x + 1)}^{2} - {(x - 2)}^{2} = {(x + 3)}^{2}$

Step 1.2

Subtract ${(x + 3)}^{2}$ from both sides of the equation.

${(x - 3)}^{2} + {(x + 1)}^{2} - {(x - 2)}^{2} - {(x + 3)}^{2} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Regroup terms.

${(x - 3)}^{2} - {(x + 3)}^{2} + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x - 3$ and $b = x + 3$ .

$(x - 3 + x + 3) (x - 3 - (x + 3)) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3

Simplify.

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Step 2.3.1

Add $x$ and $x$ .

$(2 x - 3 + 3) (x - 3 - (x + 3)) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.2

Add $- 3$ and $3$ .

$(2 x + 0) (x - 3 - (x + 3)) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.3

Add $2 x$ and $0$ .

$2 x (x - 3 - (x + 3)) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.4

Apply the distributive property.

$2 x (x - 3 - x - 1 \cdot 3) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.5

Multiply $- 1$ by $3$ .

$2 x (x - 3 - x - 3) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.6

Subtract $x$ from $x$ .

$2 x (0 - 3 - 3) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.7

Subtract $3$ from $0$ .

$2 x (- 3 - 3) + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.3.8

Subtract $3$ from $- 3$ .

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

Step 2.3.9

Multiply $- 6$ by $2$ .

$- 12 x + {(x + 1)}^{2} - {(x - 2)}^{2} = 0$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + 1$ and $b = x - 2$ .

$- 12 x + (x + 1 + x - 2) (x + 1 - (x - 2)) = 0$

Step 2.5

Simplify.

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Step 2.5.1

Add $x$ and $x$ .

$- 12 x + (2 x + 1 - 2) (x + 1 - (x - 2)) = 0$

Step 2.5.2

Subtract $2$ from $1$ .

$- 12 x + (2 x - 1) (x + 1 - (x - 2)) = 0$

Step 2.5.3

Apply the distributive property.

$- 12 x + (2 x - 1) (x + 1 - x + 2) = 0$

Step 2.5.4

Multiply $- 1$ by $- 2$ .

$- 12 x + (2 x - 1) (x + 1 - x + 2) = 0$

Step 2.5.5

Subtract $x$ from $x$ .

$- 12 x + (2 x - 1) (0 + 1 + 2) = 0$

Step 2.5.6

Add $0$ and $1$ .

$- 12 x + (2 x - 1) (1 + 2) = 0$

Step 2.5.7

Add $1$ and $2$ .

$- 12 x + (2 x - 1) \cdot 3 = 0$

Step 2.6

Factor $3$ out of $- 12 x + (2 x - 1) \cdot 3$ .

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Step 2.6.1

Factor $3$ out of $- 12 x$ .

$3 (- 4 x) + (2 x - 1) \cdot 3 = 0$

Step 2.6.2

Factor $3$ out of $(2 x - 1) \cdot 3$ .

$3 (- 4 x) + 3 \cdot (2 x - 1) = 0$

Step 2.6.3

Factor $3$ out of $3 (- 4 x) + 3 \cdot (2 x - 1)$ .

$3 (- 4 x + 2 x - 1) = 0$

Step 2.7

Add $- 4 x$ and $2 x$ .

$3 (- 2 x - 1) = 0$

Step 3

Divide each term in $3 (- 2 x - 1) = 0$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 (- 2 x - 1) = 0$ by $3$ .

$\frac{3 (- 2 x - 1)}{3} = \frac{0}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 (- 2 x - 1)}{3} = \frac{0}{3}$

Step 3.2.1.2

Divide $- 2 x - 1$ by $1$ .

$- 2 x - 1 = \frac{0}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $3$ .

$- 2 x - 1 = 0$

Step 4

Add $1$ to both sides of the equation.

$- 2 x = 1$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.5$