Algebra Examples

$\frac{{(x + 2)}^{2}}{5} + \frac{{(x - 2)}^{2}}{3} = \frac{16}{3}$

Step 1

Subtract $\frac{16}{3}$ from both sides of the equation.

$\frac{{(x + 2)}^{2}}{5} + \frac{{(x - 2)}^{2}}{3} - \frac{16}{3} = 0$

Step 2

Multiply through by the least common denominator $15$ , then simplify.

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Step 2.1

To write $\frac{{(x + 2)}^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$15 (\frac{{(x + 2)}^{2}}{5} \cdot \frac{3}{3} + \frac{{(x - 2)}^{2}}{3} - \frac{16}{3}) = 0$

Step 2.2

To write $\frac{{(x - 2)}^{2}}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$15 (\frac{{(x + 2)}^{2}}{5} \cdot \frac{3}{3} + \frac{{(x - 2)}^{2}}{3} \cdot \frac{5}{5} - \frac{16}{3}) = 0$

Step 2.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{{(x + 2)}^{2}}{5}$ by $\frac{3}{3}$ .

$15 (\frac{{(x + 2)}^{2} \cdot 3}{5 \cdot 3} + \frac{{(x - 2)}^{2}}{3} \cdot \frac{5}{5} - \frac{16}{3}) = 0$

Step 2.3.2

Multiply $5$ by $3$ .

$15 (\frac{{(x + 2)}^{2} \cdot 3}{15} + \frac{{(x - 2)}^{2}}{3} \cdot \frac{5}{5} - \frac{16}{3}) = 0$

Step 2.3.3

Multiply $\frac{{(x - 2)}^{2}}{3}$ by $\frac{5}{5}$ .

$15 (\frac{{(x + 2)}^{2} \cdot 3}{15} + \frac{{(x - 2)}^{2} \cdot 5}{3 \cdot 5} - \frac{16}{3}) = 0$

Step 2.3.4

Multiply $3$ by $5$ .

$15 (\frac{{(x + 2)}^{2} \cdot 3}{15} + \frac{{(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.4

Combine the numerators over the common denominator.

$15 (\frac{{(x + 2)}^{2} \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$15 (\frac{(x + 2) (x + 2) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.5.2.1

Apply the distributive property.

$15 (\frac{(x (x + 2) + 2 (x + 2)) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.2.2

Apply the distributive property.

$15 (\frac{(x \cdot x + x \cdot 2 + 2 (x + 2)) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.2.3

Apply the distributive property.

$15 (\frac{(x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.3

Simplify and combine like terms.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Multiply $x$ by $x$ .

$15 (\frac{(x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.3.1.2

Move $2$ to the left of $x$ .

$15 (\frac{(x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.3.1.3

Multiply $2$ by $2$ .

$15 (\frac{(x^{2} + 2 x + 2 x + 4) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.3.2

Add $2 x$ and $2 x$ .

$15 (\frac{(x^{2} + 4 x + 4) \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.4

Apply the distributive property.

$15 (\frac{x^{2} \cdot 3 + 4 x \cdot 3 + 4 \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.5

Simplify.

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Step 2.5.5.1

Move $3$ to the left of $x^{2}$ .

$15 (\frac{3 \cdot x^{2} + 4 x \cdot 3 + 4 \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.5.2

Multiply $3$ by $4$ .

$15 (\frac{3 \cdot x^{2} + 12 x + 4 \cdot 3 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.5.3

Multiply $4$ by $3$ .

$15 (\frac{3 \cdot x^{2} + 12 x + 12 + {(x - 2)}^{2} \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.6

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$15 (\frac{3 x^{2} + 12 x + 12 + (x - 2) (x - 2) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.7

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

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Step 2.5.7.1

Apply the distributive property.

$15 (\frac{3 x^{2} + 12 x + 12 + (x (x - 2) - 2 (x - 2)) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.7.2

Apply the distributive property.

$15 (\frac{3 x^{2} + 12 x + 12 + (x \cdot x + x \cdot - 2 - 2 (x - 2)) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.7.3

Apply the distributive property.

$15 (\frac{3 x^{2} + 12 x + 12 + (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.8

Simplify and combine like terms.

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Step 2.5.8.1

Simplify each term.

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Step 2.5.8.1.1

Multiply $x$ by $x$ .

$15 (\frac{3 x^{2} + 12 x + 12 + (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.8.1.2

Move $- 2$ to the left of $x$ .

$15 (\frac{3 x^{2} + 12 x + 12 + (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.8.1.3

Multiply $- 2$ by $- 2$ .

$15 (\frac{3 x^{2} + 12 x + 12 + (x^{2} - 2 x - 2 x + 4) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.8.2

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

$15 (\frac{3 x^{2} + 12 x + 12 + (x^{2} - 4 x + 4) \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.9

Apply the distributive property.

$15 (\frac{3 x^{2} + 12 x + 12 + x^{2} \cdot 5 - 4 x \cdot 5 + 4 \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.10

Simplify.

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Step 2.5.10.1

Move $5$ to the left of $x^{2}$ .

$15 (\frac{3 x^{2} + 12 x + 12 + 5 \cdot x^{2} - 4 x \cdot 5 + 4 \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.10.2

Multiply $5$ by $- 4$ .

$15 (\frac{3 x^{2} + 12 x + 12 + 5 \cdot x^{2} - 20 x + 4 \cdot 5}{15} - \frac{16}{3}) = 0$

Step 2.5.10.3

Multiply $4$ by $5$ .

$15 (\frac{3 x^{2} + 12 x + 12 + 5 \cdot x^{2} - 20 x + 20}{15} - \frac{16}{3}) = 0$

Step 2.5.11

Add $3 x^{2}$ and $5 x^{2}$ .

$15 (\frac{8 x^{2} + 12 x + 12 - 20 x + 20}{15} - \frac{16}{3}) = 0$

Step 2.5.12

Subtract $20 x$ from $12 x$ .

$15 (\frac{8 x^{2} - 8 x + 12 + 20}{15} - \frac{16}{3}) = 0$

Step 2.5.13

Add $12$ and $20$ .

$15 (\frac{8 x^{2} - 8 x + 32}{15} - \frac{16}{3}) = 0$

Step 2.5.14

Factor $8$ out of $8 x^{2} - 8 x + 32$ .

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Step 2.5.14.1

Factor $8$ out of $8 x^{2}$ .

$15 (\frac{8 (x^{2}) - 8 x + 32}{15} - \frac{16}{3}) = 0$

Step 2.5.14.2

Factor $8$ out of $- 8 x$ .

$15 (\frac{8 (x^{2}) + 8 (- x) + 32}{15} - \frac{16}{3}) = 0$

Step 2.5.14.3

Factor $8$ out of $32$ .

$15 (\frac{8 (x^{2}) + 8 (- x) + 8 (4)}{15} - \frac{16}{3}) = 0$

Step 2.5.14.4

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

$15 (\frac{8 (x^{2} - x) + 8 (4)}{15} - \frac{16}{3}) = 0$

Step 2.5.14.5

Factor $8$ out of $8 (x^{2} - x) + 8 (4)$ .

$15 (\frac{8 (x^{2} - x + 4)}{15} - \frac{16}{3}) = 0$

Step 2.6

To write $- \frac{16}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$15 (\frac{8 (x^{2} - x + 4)}{15} - \frac{16}{3} \cdot \frac{5}{5}) = 0$

Step 2.7

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{16}{3}$ by $\frac{5}{5}$ .

$15 (\frac{8 (x^{2} - x + 4)}{15} - \frac{16 \cdot 5}{3 \cdot 5}) = 0$

Step 2.7.2

Multiply $3$ by $5$ .

$15 (\frac{8 (x^{2} - x + 4)}{15} - \frac{16 \cdot 5}{15}) = 0$

Step 2.8

Combine the numerators over the common denominator.

$15 (\frac{8 (x^{2} - x + 4) - 16 \cdot 5}{15}) = 0$

Step 2.9

Cancel the common factor of $15$ .

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Step 2.9.1

Cancel the common factor.

$15 (\frac{8 (x^{2} - x + 4) - 16 \cdot 5}{15}) = 0$

Step 2.9.2

Rewrite the expression.

$8 (x^{2} - x + 4) - 16 \cdot 5 = 0$

Step 2.10

Simplify each term.

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Step 2.10.1

Apply the distributive property.

$8 x^{2} + 8 (- x) + 8 \cdot 4 - 16 \cdot 5 = 0$

Step 2.10.2

Simplify.

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Step 2.10.2.1

Multiply $- 1$ by $8$ .

$8 x^{2} - 8 x + 8 \cdot 4 - 16 \cdot 5 = 0$

Step 2.10.2.2

Multiply $8$ by $4$ .

$8 x^{2} - 8 x + 32 - 16 \cdot 5 = 0$

Step 2.10.3

Multiply $- 16$ by $5$ .

$8 x^{2} - 8 x + 32 - 80 = 0$

Step 2.11

Subtract $80$ from $32$ .

$8 x^{2} - 8 x - 48 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 8$ , $b = - 8$ , and $c = - 48$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (8 \cdot - 48)}}{2 \cdot 8}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 8 \cdot - 48}}{2 \cdot 8}$

Step 5.1.2

Multiply $- 4 \cdot 8 \cdot - 48$ .

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Step 5.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{8 \pm \sqrt{64 - 32 \cdot - 48}}{2 \cdot 8}$

Step 5.1.2.2

Multiply $- 32$ by $- 48$ .

$x = \frac{8 \pm \sqrt{64 + 1536}}{2 \cdot 8}$

Step 5.1.3

Add $64$ and $1536$ .

$x = \frac{8 \pm \sqrt{1600}}{2 \cdot 8}$

Step 5.1.4

Rewrite $1600$ as $40^{2}$ .

$x = \frac{8 \pm \sqrt{40^{2}}}{2 \cdot 8}$

Step 5.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{8 \pm 40}{2 \cdot 8}$

Step 5.2

Multiply $2$ by $8$ .

$x = \frac{8 \pm 40}{16}$

Step 5.3

Simplify $\frac{8 \pm 40}{16}$ .

$x = \frac{1 \pm 5}{2}$

Step 6

The final answer is the combination of both solutions.

$x = 3, - 2$