Solve (sqrt{20}+a)(bsqrt{5}+3)=-11

$$( \sqrt { 20 } + a ) ( b \sqrt { 5 } + 3 ) = - 11$$

Solve for a

$$a=-\frac{10b+6\sqrt{5}+11}{\sqrt{5}b+3}$$

$$b\neq -\frac{3\sqrt{5}}{5}$$

Steps for Solving Linear Equation

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\left(2\sqrt{5}+a\right)\left(b\sqrt{5}+3\right)=-11$$

Use the distributive property to multiply 2\sqrt{5}+a by b\sqrt{5}+3.

$$2b\left(\sqrt{5}\right)^{2}+6\sqrt{5}+ab\sqrt{5}+3a=-11$$

The square of \sqrt{5} is 5.

$$2b\times 5+6\sqrt{5}+ab\sqrt{5}+3a=-11$$

Multiply 2 and 5 to get 10.

$$10b+6\sqrt{5}+ab\sqrt{5}+3a=-11$$

Subtract 10b from both sides.

$$6\sqrt{5}+ab\sqrt{5}+3a=-11-10b$$

Subtract 6\sqrt{5} from both sides.

$$ab\sqrt{5}+3a=-11-10b-6\sqrt{5}$$

Combine all terms containing a.

$$\left(b\sqrt{5}+3\right)a=-11-10b-6\sqrt{5}$$

The equation is in standard form.

$$\left(\sqrt{5}b+3\right)a=-10b-6\sqrt{5}-11$$

Divide both sides by b\sqrt{5}+3.

$$\frac{\left(\sqrt{5}b+3\right)a}{\sqrt{5}b+3}=\frac{-10b-6\sqrt{5}-11}{\sqrt{5}b+3}$$

Dividing by b\sqrt{5}+3 undoes the multiplication by b\sqrt{5}+3.

$$a=\frac{-10b-6\sqrt{5}-11}{\sqrt{5}b+3}$$

Divide -11-10b-6\sqrt{5} by b\sqrt{5}+3.

$$a=-\frac{10b+6\sqrt{5}+11}{\sqrt{5}b+3}$$

Solve for b

$$b=-\frac{3a+6\sqrt{5}+11}{\sqrt{5}a+10}$$

$$a\neq -2\sqrt{5}$$

Steps for Solving Linear Equation

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\left(2\sqrt{5}+a\right)\left(b\sqrt{5}+3\right)=-11$$

Use the distributive property to multiply 2\sqrt{5}+a by b\sqrt{5}+3.

$$2b\left(\sqrt{5}\right)^{2}+6\sqrt{5}+ab\sqrt{5}+3a=-11$$

The square of \sqrt{5} is 5.

$$2b\times 5+6\sqrt{5}+ab\sqrt{5}+3a=-11$$

Multiply 2 and 5 to get 10.

$$10b+6\sqrt{5}+ab\sqrt{5}+3a=-11$$

Subtract 6\sqrt{5} from both sides.

$$10b+ab\sqrt{5}+3a=-11-6\sqrt{5}$$

Subtract 3a from both sides.

$$10b+ab\sqrt{5}=-11-6\sqrt{5}-3a$$

Combine all terms containing b.

$$\left(10+a\sqrt{5}\right)b=-11-6\sqrt{5}-3a$$

The equation is in standard form.

$$\left(\sqrt{5}a+10\right)b=-3a-6\sqrt{5}-11$$

Divide both sides by 10+a\sqrt{5}.

$$\frac{\left(\sqrt{5}a+10\right)b}{\sqrt{5}a+10}=\frac{-3a-6\sqrt{5}-11}{\sqrt{5}a+10}$$

Dividing by 10+a\sqrt{5} undoes the multiplication by 10+a\sqrt{5}.

$$b=\frac{-3a-6\sqrt{5}-11}{\sqrt{5}a+10}$$

Divide -11-6\sqrt{5}-3a by 10+a\sqrt{5}.

$$b=-\frac{3a+6\sqrt{5}+11}{\sqrt{5}a+10}$$