Solve sqrt{4x+3}=2sqrt{x-1}+1

$$\sqrt { 4 x + 3 } = 2 \sqrt { x - 1 } + 1$$

Solve for x

$$x = \frac{13}{4} = 3\frac{1}{4} = 3.25$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4x+3}\right)^{2}=\left(2\sqrt{x-1}+1\right)^{2}$$

Calculate \sqrt{4x+3} to the power of 2 and get 4x+3.

$$4x+3=\left(2\sqrt{x-1}+1\right)^{2}$$

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

$$4x+3=4\left(\sqrt{x-1}\right)^{2}+4\sqrt{x-1}+1$$

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

$$4x+3=4\left(x-1\right)+4\sqrt{x-1}+1$$

Use the distributive property to multiply 4 by x-1.

$$4x+3=4x-4+4\sqrt{x-1}+1$$

Add -4 and 1 to get -3.

$$4x+3=4x-3+4\sqrt{x-1}$$

Subtract 4x from both sides.

$$4x+3-4x=-3+4\sqrt{x-1}$$

Combine 4x and -4x to get 0.

$$3=-3+4\sqrt{x-1}$$

Swap sides so that all variable terms are on the left hand side.

$$-3+4\sqrt{x-1}=3$$

Add 3 to both sides.

$$4\sqrt{x-1}=3+3$$

Add 3 and 3 to get 6.

$$4\sqrt{x-1}=6$$

Divide both sides by 4.

$$\sqrt{x-1}=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

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

Square both sides of the equation.

$$x-1=\frac{9}{4}$$

Add 1 to both sides of the equation.

$$x-1-\left(-1\right)=\frac{9}{4}-\left(-1\right)$$

Subtracting -1 from itself leaves 0.

$$x=\frac{9}{4}-\left(-1\right)$$

Subtract -1 from \frac{9}{4}.

$$x=\frac{13}{4}$$

Substitute \frac{13}{4} for x in the equation \sqrt{4x+3}=2\sqrt{x-1}+1.

$$\sqrt{4\times \frac{13}{4}+3}=2\sqrt{\frac{13}{4}-1}+1$$

Simplify. The value x=\frac{13}{4} satisfies the equation.

$$4=4$$

Equation \sqrt{4x+3}=2\sqrt{x-1}+1 has a unique solution.

$$x=\frac{13}{4}$$