Algebra Examples

${(x - 5)}^{2} - x^{2} = 3$

Step 1

Subtract $3$ from both sides of the equation.

${(x - 5)}^{2} - x^{2} - 3 = 0$

Step 2

Simplify ${(x - 5)}^{2} - x^{2} - 3$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$(x - 5) (x - 5) - x^{2} - 3 = 0$

Step 2.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x (x - 5) - 5 (x - 5) - x^{2} - 3 = 0$

Step 2.1.2.2

Apply the distributive property.

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

Step 2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - x^{2} - 3 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - x^{2} - 3 = 0$

Step 2.1.3.1.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - x^{2} - 3 = 0$

Step 2.1.3.1.3

Multiply $- 5$ by $- 5$ .

$x^{2} - 5 x - 5 x + 25 - x^{2} - 3 = 0$

Step 2.1.3.2

Subtract $5 x$ from $- 5 x$ .

$x^{2} - 10 x + 25 - x^{2} - 3 = 0$

Step 2.2

Combine the opposite terms in $x^{2} - 10 x + 25 - x^{2} - 3$ .

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Step 2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- 10 x + 25 + 0 - 3 = 0$

Step 2.2.2

Add $- 10 x + 25$ and $0$ .

$- 10 x + 25 - 3 = 0$

Step 2.3

Subtract $3$ from $25$ .

$- 10 x + 22 = 0$

Step 3

Factor $- 2$ out of $- 10 x + 22$ .

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Step 3.1

Factor $- 2$ out of $- 10 x$ .

$- 2 (5 x) + 22 = 0$

Step 3.2

Factor $- 2$ out of $22$ .

$- 2 (5 x) - 2 \cdot - 11 = 0$

Step 3.3

Factor $- 2$ out of $- 2 (5 x) - 2 (- 11)$ .

$- 2 (5 x - 11) = 0$

Step 4

Divide each term in $- 2 (5 x - 11) = 0$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 (5 x - 11) = 0$ by $- 2$ .

$\frac{- 2 (5 x - 11)}{- 2} = \frac{0}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 (5 x - 11)}{- 2} = \frac{0}{- 2}$

Step 4.2.1.2

Divide $5 x - 11$ by $1$ .

$5 x - 11 = \frac{0}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 2$ .

$5 x - 11 = 0$

Step 5

Add $11$ to both sides of the equation.

$5 x = 11$

Step 6

Divide each term in $5 x = 11$ by $5$ and simplify.

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Step 6.1

Divide each term in $5 x = 11$ by $5$ .

$\frac{5 x}{5} = \frac{11}{5}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $5$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{11}{5}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{5}$

Step 7