Solve for x (x-5)^2-x^2=3

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

Step 1

Subtract $3$ from both sides of the equation.

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

Step 2

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 5$ by $- 5$ .

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

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

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

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

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

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

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

$- 10 x + 25 - 3 = 0$

Subtract $3$ from $25$ .

$- 10 x + 22 = 0$

Step 3

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

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

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

Factor $- 2$ out of $22$ .

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

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.

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

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.

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.2$

Mixed Number Form:

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

Solve for x (x-5)^2-x^2=3

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