Solve sqrt{2x+1}=3sqrt{x-1}

$$\sqrt { 2 x + 1 } = 3 \sqrt { x - 1 }$$

Solve for x

$$x = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

$$2x+1=9\left(x-1\right)$$

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

$$2x+1=9x-9$$

Subtract 9x from both sides.

$$2x+1-9x=-9$$

Combine 2x and -9x to get -7x.

$$-7x+1=-9$$

Subtract 1 from both sides.

$$-7x=-9-1$$

Subtract 1 from -9 to get -10.

$$-7x=-10$$

Divide both sides by -7.

$$x=\frac{-10}{-7}$$

Fraction \frac{-10}{-7} can be simplified to \frac{10}{7} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{10}{7}$$

Substitute \frac{10}{7} for x in the equation \sqrt{2x+1}=3\sqrt{x-1}.

$$\sqrt{2\times \frac{10}{7}+1}=3\sqrt{\frac{10}{7}-1}$$

Simplify. The value x=\frac{10}{7} satisfies the equation.

$$\frac{3}{7}\times 21^{\frac{1}{2}}=\frac{3}{7}\times 21^{\frac{1}{2}}$$

Equation \sqrt{2x+1}=3\sqrt{x-1} has a unique solution.

$$x=\frac{10}{7}$$