Pre-Algebra Examples

${(x - 2)}^{2} + {(x - 4)}^{2} = 9$

Step 1

Simplify ${(x - 2)}^{2} + {(x - 4)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$(x - 2) (x - 2) + {(x - 4)}^{2} = 9$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$x (x - 2) - 2 (x - 2) + {(x - 4)}^{2} = 9$

Step 1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) + {(x - 4)}^{2} = 9$

Step 1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + {(x - 4)}^{2} = 9$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 + {(x - 4)}^{2} = 9$

Step 1.1.3.1.2

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

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 + {(x - 4)}^{2} = 9$

Step 1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4 + {(x - 4)}^{2} = 9$

Step 1.1.3.2

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

$x^{2} - 4 x + 4 + {(x - 4)}^{2} = 9$

Step 1.1.4

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$x^{2} - 4 x + 4 + (x - 4) (x - 4) = 9$

Step 1.1.5

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

$x^{2} - 4 x + 4 + x (x - 4) - 4 (x - 4) = 9$

Step 1.1.5.2

Apply the distributive property.

$x^{2} - 4 x + 4 + x \cdot x + x \cdot - 4 - 4 (x - 4) = 9$

Step 1.1.5.3

Apply the distributive property.

$x^{2} - 4 x + 4 + x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 = 9$

Step 1.1.6

Simplify and combine like terms.

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Step 1.1.6.1

Simplify each term.

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Step 1.1.6.1.1

Multiply $x$ by $x$ .

$x^{2} - 4 x + 4 + x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 = 9$

Step 1.1.6.1.2

Move $- 4$ to the left of $x$ .

$x^{2} - 4 x + 4 + x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 = 9$

Step 1.1.6.1.3

Multiply $- 4$ by $- 4$ .

$x^{2} - 4 x + 4 + x^{2} - 4 x - 4 x + 16 = 9$

Step 1.1.6.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 4 x + 4 + x^{2} - 8 x + 16 = 9$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

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

$2 x^{2} - 4 x + 4 - 8 x + 16 = 9$

Step 1.2.2

Subtract $8 x$ from $- 4 x$ .

$2 x^{2} - 12 x + 4 + 16 = 9$

Step 1.2.3

Add $4$ and $16$ .

$2 x^{2} - 12 x + 20 = 9$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.1291713, 4.87082869$

Step 3