Solve 3sqrt{x+1}-3sqrt{x-6}=1

$$3 \sqrt{ x+1 } -3 \sqrt{ x-6 } =1$$

Solve for x

$$x = \frac{1015}{9} = 112\frac{7}{9} \approx 112.777777778$$

Solution Steps

Subtract -3\sqrt{x-6} from both sides of the equation.

$$3\sqrt{x+1}=1+3\sqrt{x-6}$$

Square both sides of the equation.

$$\left(3\sqrt{x+1}\right)^{2}=\left(1+3\sqrt{x-6}\right)^{2}$$

Expand \left(3\sqrt{x+1}\right)^{2}.

$$3^{2}\left(\sqrt{x+1}\right)^{2}=\left(1+3\sqrt{x-6}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x+1}\right)^{2}=\left(1+3\sqrt{x-6}\right)^{2}$$

Calculate \sqrt{x+1} to the power of 2 and get x+1.

$$9\left(x+1\right)=\left(1+3\sqrt{x-6}\right)^{2}$$

Use the distributive property to multiply 9 by x+1.

$$9x+9=\left(1+3\sqrt{x-6}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+3\sqrt{x-6}\right)^{2}.

$$9x+9=1+6\sqrt{x-6}+9\left(\sqrt{x-6}\right)^{2}$$

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

$$9x+9=1+6\sqrt{x-6}+9\left(x-6\right)$$

Use the distributive property to multiply 9 by x-6.

$$9x+9=1+6\sqrt{x-6}+9x-54$$

Subtract 54 from 1 to get -53.

$$9x+9=-53+6\sqrt{x-6}+9x$$

Subtract 6\sqrt{x-6} from both sides.

$$9x+9-6\sqrt{x-6}=-53+9x$$

Subtract 9x from both sides.

$$9x+9-6\sqrt{x-6}-9x=-53$$

Combine 9x and -9x to get 0.

$$9-6\sqrt{x-6}=-53$$

Subtract 9 from both sides.

$$-6\sqrt{x-6}=-53-9$$

Subtract 9 from -53 to get -62.

$$-6\sqrt{x-6}=-62$$

Divide both sides by -6.

$$\sqrt{x-6}=\frac{-62}{-6}$$

Reduce the fraction \frac{-62}{-6} to lowest terms by extracting and canceling out -2.

$$\sqrt{x-6}=\frac{31}{3}$$

Square both sides of the equation.

$$x-6=\frac{961}{9}$$

Add 6 to both sides of the equation.

$$x-6-\left(-6\right)=\frac{961}{9}-\left(-6\right)$$

Subtracting -6 from itself leaves 0.

$$x=\frac{961}{9}-\left(-6\right)$$

Subtract -6 from \frac{961}{9}.

$$x=\frac{1015}{9}$$

Substitute \frac{1015}{9} for x in the equation 3\sqrt{x+1}-3\sqrt{x-6}=1.

$$3\sqrt{\frac{1015}{9}+1}-3\sqrt{\frac{1015}{9}-6}=1$$

Simplify. The value x=\frac{1015}{9} satisfies the equation.

$$1=1$$

Equation 3\sqrt{x+1}=3\sqrt{x-6}+1 has a unique solution.

$$x=\frac{1015}{9}$$