Solve 3sqrt{8x}+sqrt{2x}=1

$$3 \sqrt { 8 x } + \sqrt { 2 x } = 1$$

Solve for x

$$x=\frac{1}{98}\approx 0.010204082$$

Solution Steps

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

$$3\sqrt{8x}=1-\sqrt{2x}$$

Square both sides of the equation.

$$\left(3\sqrt{8x}\right)^{2}=\left(1-\sqrt{2x}\right)^{2}$$

Expand \left(3\sqrt{8x}\right)^{2}.

$$3^{2}\left(\sqrt{8x}\right)^{2}=\left(1-\sqrt{2x}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{8x}\right)^{2}=\left(1-\sqrt{2x}\right)^{2}$$

Calculate \sqrt{8x} to the power of 2 and get 8x.

$$9\times 8x=\left(1-\sqrt{2x}\right)^{2}$$

Multiply 9 and 8 to get 72.

$$72x=\left(1-\sqrt{2x}\right)^{2}$$

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

$$72x=1-2\sqrt{2x}+\left(\sqrt{2x}\right)^{2}$$

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

$$72x=1-2\sqrt{2x}+2x$$

Subtract 1+2x from both sides of the equation.

$$72x-\left(1+2x\right)=-2\sqrt{2x}$$

To find the opposite of 1+2x, find the opposite of each term.

$$72x-1-2x=-2\sqrt{2x}$$

Combine 72x and -2x to get 70x.

$$70x-1=-2\sqrt{2x}$$

Square both sides of the equation.

$$\left(70x-1\right)^{2}=\left(-2\sqrt{2x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(70x-1\right)^{2}.

$$4900x^{2}-140x+1=\left(-2\sqrt{2x}\right)^{2}$$

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

$$4900x^{2}-140x+1=\left(-2\right)^{2}\left(\sqrt{2x}\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4900x^{2}-140x+1=4\left(\sqrt{2x}\right)^{2}$$

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

$$4900x^{2}-140x+1=4\times 2x$$

Multiply 4 and 2 to get 8.

$$4900x^{2}-140x+1=8x$$

Subtract 8x from both sides.

$$4900x^{2}-140x+1-8x=0$$

Combine -140x and -8x to get -148x.

$$4900x^{2}-148x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4900 for a, -148 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-148\right)±\sqrt{\left(-148\right)^{2}-4\times 4900}}{2\times 4900}$$

Square -148.

$$x=\frac{-\left(-148\right)±\sqrt{21904-4\times 4900}}{2\times 4900}$$

Multiply -4 times 4900.

$$x=\frac{-\left(-148\right)±\sqrt{21904-19600}}{2\times 4900}$$

Add 21904 to -19600.

$$x=\frac{-\left(-148\right)±\sqrt{2304}}{2\times 4900}$$

Take the square root of 2304.

$$x=\frac{-\left(-148\right)±48}{2\times 4900}$$

The opposite of -148 is 148.

$$x=\frac{148±48}{2\times 4900}$$

Multiply 2 times 4900.

$$x=\frac{148±48}{9800}$$

Now solve the equation x=\frac{148±48}{9800} when ± is plus. Add 148 to 48.

$$x=\frac{196}{9800}$$

Reduce the fraction \frac{196}{9800} to lowest terms by extracting and canceling out 196.

$$x=\frac{1}{50}$$

Now solve the equation x=\frac{148±48}{9800} when ± is minus. Subtract 48 from 148.

$$x=\frac{100}{9800}$$

Reduce the fraction \frac{100}{9800} to lowest terms by extracting and canceling out 100.

$$x=\frac{1}{98}$$

The equation is now solved.

$$x=\frac{1}{50} x=\frac{1}{98}$$

Substitute \frac{1}{50} for x in the equation 3\sqrt{8x}+\sqrt{2x}=1.

$$3\sqrt{8\times \frac{1}{50}}+\sqrt{2\times \frac{1}{50}}=1$$

Simplify. The value x=\frac{1}{50} does not satisfy the equation.

$$\frac{7}{5}=1$$

Substitute \frac{1}{98} for x in the equation 3\sqrt{8x}+\sqrt{2x}=1.

$$3\sqrt{8\times \frac{1}{98}}+\sqrt{2\times \frac{1}{98}}=1$$

Simplify. The value x=\frac{1}{98} satisfies the equation.

$$1=1$$

Equation 3\sqrt{8x}=-\sqrt{2x}+1 has a unique solution.

$$x=\frac{1}{98}$$