Solve sqrt{x+6}-sqrt{9x+70}=-2sqrt{x+9}

$$\sqrt{ x+6 } - \sqrt{ 9x+70 } = -2 \sqrt{ x+9 }$$

Solve for x

$$x=-5$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+6}-\sqrt{9x+70}\right)^{2}=\left(-2\sqrt{x+9}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{x+6}-\sqrt{9x+70}\right)^{2}.

$$\left(\sqrt{x+6}\right)^{2}-2\sqrt{x+6}\sqrt{9x+70}+\left(\sqrt{9x+70}\right)^{2}=\left(-2\sqrt{x+9}\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$x+6-2\sqrt{x+6}\sqrt{9x+70}+\left(\sqrt{9x+70}\right)^{2}=\left(-2\sqrt{x+9}\right)^{2}$$

Calculate \sqrt{9x+70} to the power of 2 and get 9x+70.

$$x+6-2\sqrt{x+6}\sqrt{9x+70}+9x+70=\left(-2\sqrt{x+9}\right)^{2}$$

Combine x and 9x to get 10x.

$$10x+6-2\sqrt{x+6}\sqrt{9x+70}+70=\left(-2\sqrt{x+9}\right)^{2}$$

Add 6 and 70 to get 76.

$$10x+76-2\sqrt{x+6}\sqrt{9x+70}=\left(-2\sqrt{x+9}\right)^{2}$$

Expand \left(-2\sqrt{x+9}\right)^{2}.

$$10x+76-2\sqrt{x+6}\sqrt{9x+70}=\left(-2\right)^{2}\left(\sqrt{x+9}\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$10x+76-2\sqrt{x+6}\sqrt{9x+70}=4\left(\sqrt{x+9}\right)^{2}$$

Calculate \sqrt{x+9} to the power of 2 and get x+9.

$$10x+76-2\sqrt{x+6}\sqrt{9x+70}=4\left(x+9\right)$$

Use the distributive property to multiply 4 by x+9.

$$10x+76-2\sqrt{x+6}\sqrt{9x+70}=4x+36$$

Subtract 10x+76 from both sides of the equation.

$$-2\sqrt{x+6}\sqrt{9x+70}=4x+36-\left(10x+76\right)$$

To find the opposite of 10x+76, find the opposite of each term.

$$-2\sqrt{x+6}\sqrt{9x+70}=4x+36-10x-76$$

Combine 4x and -10x to get -6x.

$$-2\sqrt{x+6}\sqrt{9x+70}=-6x+36-76$$

Subtract 76 from 36 to get -40.

$$-2\sqrt{x+6}\sqrt{9x+70}=-6x-40$$

Square both sides of the equation.

$$\left(-2\sqrt{x+6}\sqrt{9x+70}\right)^{2}=\left(-6x-40\right)^{2}$$

Expand \left(-2\sqrt{x+6}\sqrt{9x+70}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{x+6}\right)^{2}\left(\sqrt{9x+70}\right)^{2}=\left(-6x-40\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4\left(\sqrt{x+6}\right)^{2}\left(\sqrt{9x+70}\right)^{2}=\left(-6x-40\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$4\left(x+6\right)\left(\sqrt{9x+70}\right)^{2}=\left(-6x-40\right)^{2}$$

Calculate \sqrt{9x+70} to the power of 2 and get 9x+70.

$$4\left(x+6\right)\left(9x+70\right)=\left(-6x-40\right)^{2}$$

Use the distributive property to multiply 4 by x+6.

$$\left(4x+24\right)\left(9x+70\right)=\left(-6x-40\right)^{2}$$

Apply the distributive property by multiplying each term of 4x+24 by each term of 9x+70.

$$36x^{2}+280x+216x+1680=\left(-6x-40\right)^{2}$$

Combine 280x and 216x to get 496x.

$$36x^{2}+496x+1680=\left(-6x-40\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-6x-40\right)^{2}.

$$36x^{2}+496x+1680=36x^{2}+480x+1600$$

Subtract 36x^{2} from both sides.

$$36x^{2}+496x+1680-36x^{2}=480x+1600$$

Combine 36x^{2} and -36x^{2} to get 0.

$$496x+1680=480x+1600$$

Subtract 480x from both sides.

$$496x+1680-480x=1600$$

Combine 496x and -480x to get 16x.

$$16x+1680=1600$$

Subtract 1680 from both sides.

$$16x=1600-1680$$

Subtract 1680 from 1600 to get -80.

$$16x=-80$$

Divide both sides by 16.

$$x=\frac{-80}{16}$$

Divide -80 by 16 to get -5.

$$x=-5$$

Substitute -5 for x in the equation \sqrt{x+6}-\sqrt{9x+70}=-2\sqrt{x+9}.

$$\sqrt{-5+6}-\sqrt{9\left(-5\right)+70}=-2\sqrt{-5+9}$$

Simplify. The value x=-5 satisfies the equation.

$$-4=-4$$

Equation \sqrt{x+6}-\sqrt{9x+70}=-2\sqrt{x+9} has a unique solution.

$$x=-5$$