Solve for x (x-2 square root of 5)(x+5 square root of 2)=x^2-10 square root of 10

$(x - 2 \sqrt{5}) (x + 5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Step 1

Simplify $(x - 2 \sqrt{5}) (x + 5 \sqrt{2})$ .

Rewrite.

$0 + 0 + (x - 2 \sqrt{5}) (x + 5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Simplify by adding zeros.

$(x - 2 \sqrt{5}) (x + 5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Expand $(x - 2 \sqrt{5}) (x + 5 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$x (x + 5 \sqrt{2}) - 2 \sqrt{5} (x + 5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Apply the distributive property.

$x \cdot x + x (5 \sqrt{2}) - 2 \sqrt{5} (x + 5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Apply the distributive property.

$x \cdot x + x (5 \sqrt{2}) - 2 \sqrt{5} x - 2 \sqrt{5} (5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (5 \sqrt{2}) - 2 \sqrt{5} x - 2 \sqrt{5} (5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

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

$x^{2} + 5 \cdot x \sqrt{2} - 2 \sqrt{5} x - 2 \sqrt{5} (5 \sqrt{2}) = x^{2} - 10 \sqrt{10}$

Multiply $- 2 \sqrt{5} (5 \sqrt{2})$ .

Multiply $5$ by $- 2$ .

$x^{2} + 5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{5} \sqrt{2} = x^{2} - 10 \sqrt{10}$

Combine using the product rule for radicals.

$x^{2} + 5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{2 \cdot 5} = x^{2} - 10 \sqrt{10}$

Multiply $2$ by $5$ .

$x^{2} + 5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{10} = x^{2} - 10 \sqrt{10}$

Step 2

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

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

$x^{2} + 5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{10} - x^{2} = - 10 \sqrt{10}$

Combine the opposite terms in $x^{2} + 5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{10} - x^{2}$ .

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

$5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{10} + 0 = - 10 \sqrt{10}$

Add $5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{10}$ and $0$ .

$5 x \sqrt{2} - 2 \sqrt{5} x - 10 \sqrt{10} = - 10 \sqrt{10}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

Add $10 \sqrt{10}$ to both sides of the equation.

$5 x \sqrt{2} - 2 \sqrt{5} x = - 10 \sqrt{10} + 10 \sqrt{10}$

Add $- 10 \sqrt{10}$ and $10 \sqrt{10}$ .

$5 x \sqrt{2} - 2 \sqrt{5} x = 0$

Step 4

Factor $x$ out of $5 x \sqrt{2} - 2 \sqrt{5} x$ .

Factor $x$ out of $5 x \sqrt{2}$ .

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

Factor $x$ out of $- 2 \sqrt{5} x$ .

$x (5 \sqrt{2}) + x (- 2 \sqrt{5}) = 0$

Factor $x$ out of $x (5 \sqrt{2}) + x (- 2 \sqrt{5})$ .

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

Step 5

Divide each term in $x (5 \sqrt{2} - 2 \sqrt{5}) = 0$ by $5 \sqrt{2} - 2 \sqrt{5}$ and simplify.

Divide each term in $x (5 \sqrt{2} - 2 \sqrt{5}) = 0$ by $5 \sqrt{2} - 2 \sqrt{5}$ .

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

Simplify the left side.

Cancel the common factor of $5 \sqrt{2} - 2 \sqrt{5}$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $0$ by $5 \sqrt{2} - 2 \sqrt{5}$ .

$x = 0$

Solve for x (x-2 square root of 5)(x+5 square root of 2)=x^2-10 square root of 10

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