Solve 2sqrt{x-2}+4sqrt{2x-1}=8

$$2 \sqrt { x - 2 } + 4 \sqrt { 2 x - 1 } = 8$$

Solve for x

$$x = \frac{158 - 16 \sqrt{11}}{49} \approx 2.141510273$$

Solution Steps

Subtract 4\sqrt{2x-1} from both sides of the equation.

$$2\sqrt{x-2}=8-4\sqrt{2x-1}$$

Square both sides of the equation.

$$\left(2\sqrt{x-2}\right)^{2}=\left(8-4\sqrt{2x-1}\right)^{2}$$

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

$$2^{2}\left(\sqrt{x-2}\right)^{2}=\left(8-4\sqrt{2x-1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x-2}\right)^{2}=\left(8-4\sqrt{2x-1}\right)^{2}$$

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

$$4\left(x-2\right)=\left(8-4\sqrt{2x-1}\right)^{2}$$

Use the distributive property to multiply 4 by x-2.

$$4x-8=\left(8-4\sqrt{2x-1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-4\sqrt{2x-1}\right)^{2}.

$$4x-8=64-64\sqrt{2x-1}+16\left(\sqrt{2x-1}\right)^{2}$$

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

$$4x-8=64-64\sqrt{2x-1}+16\left(2x-1\right)$$

Use the distributive property to multiply 16 by 2x-1.

$$4x-8=64-64\sqrt{2x-1}+32x-16$$

Subtract 16 from 64 to get 48.

$$4x-8=48-64\sqrt{2x-1}+32x$$

Subtract 48+32x from both sides of the equation.

$$4x-8-\left(48+32x\right)=-64\sqrt{2x-1}$$

To find the opposite of 48+32x, find the opposite of each term.

$$4x-8-48-32x=-64\sqrt{2x-1}$$

Subtract 48 from -8 to get -56.

$$4x-56-32x=-64\sqrt{2x-1}$$

Combine 4x and -32x to get -28x.

$$-28x-56=-64\sqrt{2x-1}$$

Square both sides of the equation.

$$\left(-28x-56\right)^{2}=\left(-64\sqrt{2x-1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-28x-56\right)^{2}.

$$784x^{2}+3136x+3136=\left(-64\sqrt{2x-1}\right)^{2}$$

Expand \left(-64\sqrt{2x-1}\right)^{2}.

$$784x^{2}+3136x+3136=\left(-64\right)^{2}\left(\sqrt{2x-1}\right)^{2}$$

Calculate -64 to the power of 2 and get 4096.

$$784x^{2}+3136x+3136=4096\left(\sqrt{2x-1}\right)^{2}$$

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

$$784x^{2}+3136x+3136=4096\left(2x-1\right)$$

Use the distributive property to multiply 4096 by 2x-1.

$$784x^{2}+3136x+3136=8192x-4096$$

Subtract 8192x from both sides.

$$784x^{2}+3136x+3136-8192x=-4096$$

Combine 3136x and -8192x to get -5056x.

$$784x^{2}-5056x+3136=-4096$$

Add 4096 to both sides.

$$784x^{2}-5056x+3136+4096=0$$

Add 3136 and 4096 to get 7232.

$$784x^{2}-5056x+7232=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 784 for a, -5056 for b, and 7232 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5056\right)±\sqrt{\left(-5056\right)^{2}-4\times 784\times 7232}}{2\times 784}$$

Square -5056.

$$x=\frac{-\left(-5056\right)±\sqrt{25563136-4\times 784\times 7232}}{2\times 784}$$

Multiply -4 times 784.

$$x=\frac{-\left(-5056\right)±\sqrt{25563136-3136\times 7232}}{2\times 784}$$

Multiply -3136 times 7232.

$$x=\frac{-\left(-5056\right)±\sqrt{25563136-22679552}}{2\times 784}$$

Add 25563136 to -22679552.

$$x=\frac{-\left(-5056\right)±\sqrt{2883584}}{2\times 784}$$

Take the square root of 2883584.

$$x=\frac{-\left(-5056\right)±512\sqrt{11}}{2\times 784}$$

The opposite of -5056 is 5056.

$$x=\frac{5056±512\sqrt{11}}{2\times 784}$$

Multiply 2 times 784.

$$x=\frac{5056±512\sqrt{11}}{1568}$$

Now solve the equation x=\frac{5056±512\sqrt{11}}{1568} when ± is plus. Add 5056 to 512\sqrt{11}.

$$x=\frac{512\sqrt{11}+5056}{1568}$$

Divide 5056+512\sqrt{11} by 1568.

$$x=\frac{16\sqrt{11}+158}{49}$$

Now solve the equation x=\frac{5056±512\sqrt{11}}{1568} when ± is minus. Subtract 512\sqrt{11} from 5056.

$$x=\frac{5056-512\sqrt{11}}{1568}$$

Divide 5056-512\sqrt{11} by 1568.

$$x=\frac{158-16\sqrt{11}}{49}$$

The equation is now solved.

$$x=\frac{16\sqrt{11}+158}{49} x=\frac{158-16\sqrt{11}}{49}$$

Substitute \frac{16\sqrt{11}+158}{49} for x in the equation 2\sqrt{x-2}+4\sqrt{2x-1}=8.

$$2\sqrt{\frac{16\sqrt{11}+158}{49}-2}+4\sqrt{2\times \frac{16\sqrt{11}+158}{49}-1}=8$$

Simplify. The value x=\frac{16\sqrt{11}+158}{49} does not satisfy the equation.

$$\frac{8}{7}\times 11^{\frac{1}{2}}+\frac{72}{7}=8$$

Substitute \frac{158-16\sqrt{11}}{49} for x in the equation 2\sqrt{x-2}+4\sqrt{2x-1}=8.

$$2\sqrt{\frac{158-16\sqrt{11}}{49}-2}+4\sqrt{2\times \frac{158-16\sqrt{11}}{49}-1}=8$$

Simplify. The value x=\frac{158-16\sqrt{11}}{49} satisfies the equation.

$$8=8$$

Equation 2\sqrt{x-2}=-4\sqrt{2x-1}+8 has a unique solution.

$$x=\frac{158-16\sqrt{11}}{49}$$