Solve 2sqrt{4(t-1)}=sqrt{4(2t-1)}

$$2 \sqrt { 4 ( t - 1 ) } = \sqrt { 4 ( 2 t - 1 ) }$$

Solve for t

$$t = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Solution Steps

Square both sides of the equation.

$$\left(2\sqrt{4\left(t-1\right)}\right)^{2}=\left(\sqrt{4\left(2t-1\right)}\right)^{2}$$

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

$$\left(2\sqrt{4t-4}\right)^{2}=\left(\sqrt{4\left(2t-1\right)}\right)^{2}$$

Expand \left(2\sqrt{4t-4}\right)^{2}.

$$2^{2}\left(\sqrt{4t-4}\right)^{2}=\left(\sqrt{4\left(2t-1\right)}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{4t-4}\right)^{2}=\left(\sqrt{4\left(2t-1\right)}\right)^{2}$$

Calculate \sqrt{4t-4} to the power of 2 and get 4t-4.

$$4\left(4t-4\right)=\left(\sqrt{4\left(2t-1\right)}\right)^{2}$$

Use the distributive property to multiply 4 by 4t-4.

$$16t-16=\left(\sqrt{4\left(2t-1\right)}\right)^{2}$$

Use the distributive property to multiply 4 by 2t-1.

$$16t-16=\left(\sqrt{8t-4}\right)^{2}$$

Calculate \sqrt{8t-4} to the power of 2 and get 8t-4.

$$16t-16=8t-4$$

Subtract 8t from both sides.

$$16t-16-8t=-4$$

Combine 16t and -8t to get 8t.

$$8t-16=-4$$

Add 16 to both sides.

$$8t=-4+16$$

Add -4 and 16 to get 12.

$$8t=12$$

Divide both sides by 8.

$$t=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

$$t=\frac{3}{2}$$

Substitute \frac{3}{2} for t in the equation 2\sqrt{4\left(t-1\right)}=\sqrt{4\left(2t-1\right)}.

$$2\sqrt{4\left(\frac{3}{2}-1\right)}=\sqrt{4\left(2\times \frac{3}{2}-1\right)}$$

Simplify. The value t=\frac{3}{2} satisfies the equation.

$$2\times 2^{\frac{1}{2}}=2\times 2^{\frac{1}{2}}$$

Equation 2\sqrt{4\left(t-1\right)}=\sqrt{4\left(2t-1\right)} has a unique solution.

$$t=\frac{3}{2}$$