Solve 8x+140sqrt{x}=4

$$8 x + 140 \sqrt { x } = 4$$

Solve for x

$$x=\frac{1229-105\sqrt{137}}{8}\approx 0.000813672$$

Solution Steps

Subtract 8x from both sides of the equation.

$$140\sqrt{x}=4-8x$$

Square both sides of the equation.

$$\left(140\sqrt{x}\right)^{2}=\left(4-8x\right)^{2}$$

Expand \left(140\sqrt{x}\right)^{2}.

$$140^{2}\left(\sqrt{x}\right)^{2}=\left(4-8x\right)^{2}$$

Calculate 140 to the power of 2 and get 19600.

$$19600\left(\sqrt{x}\right)^{2}=\left(4-8x\right)^{2}$$

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

$$19600x=\left(4-8x\right)^{2}$$

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

$$19600x=16-64x+64x^{2}$$

Subtract 16 from both sides.

$$19600x-16=-64x+64x^{2}$$

Add 64x to both sides.

$$19600x-16+64x=64x^{2}$$

Combine 19600x and 64x to get 19664x.

$$19664x-16=64x^{2}$$

Subtract 64x^{2} from both sides.

$$19664x-16-64x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-64x^{2}+19664x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -64 for a, 19664 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19664±\sqrt{19664^{2}-4\left(-64\right)\left(-16\right)}}{2\left(-64\right)}$$

Square 19664.

$$x=\frac{-19664±\sqrt{386672896-4\left(-64\right)\left(-16\right)}}{2\left(-64\right)}$$

Multiply -4 times -64.

$$x=\frac{-19664±\sqrt{386672896+256\left(-16\right)}}{2\left(-64\right)}$$

Multiply 256 times -16.

$$x=\frac{-19664±\sqrt{386672896-4096}}{2\left(-64\right)}$$

Add 386672896 to -4096.

$$x=\frac{-19664±\sqrt{386668800}}{2\left(-64\right)}$$

Take the square root of 386668800.

$$x=\frac{-19664±1680\sqrt{137}}{2\left(-64\right)}$$

Multiply 2 times -64.

$$x=\frac{-19664±1680\sqrt{137}}{-128}$$

Now solve the equation x=\frac{-19664±1680\sqrt{137}}{-128} when ± is plus. Add -19664 to 1680\sqrt{137}.

$$x=\frac{1680\sqrt{137}-19664}{-128}$$

Divide -19664+1680\sqrt{137} by -128.

$$x=\frac{1229-105\sqrt{137}}{8}$$

Now solve the equation x=\frac{-19664±1680\sqrt{137}}{-128} when ± is minus. Subtract 1680\sqrt{137} from -19664.

$$x=\frac{-1680\sqrt{137}-19664}{-128}$$

Divide -19664-1680\sqrt{137} by -128.

$$x=\frac{105\sqrt{137}+1229}{8}$$

The equation is now solved.

$$x=\frac{1229-105\sqrt{137}}{8} x=\frac{105\sqrt{137}+1229}{8}$$

Substitute \frac{1229-105\sqrt{137}}{8} for x in the equation 8x+140\sqrt{x}=4.

$$8\times \frac{1229-105\sqrt{137}}{8}+140\sqrt{\frac{1229-105\sqrt{137}}{8}}=4$$

Simplify. The value x=\frac{1229-105\sqrt{137}}{8} satisfies the equation.

$$4=4$$

Substitute \frac{105\sqrt{137}+1229}{8} for x in the equation 8x+140\sqrt{x}=4.

$$8\times \frac{105\sqrt{137}+1229}{8}+140\sqrt{\frac{105\sqrt{137}+1229}{8}}=4$$

Simplify. The value x=\frac{105\sqrt{137}+1229}{8} does not satisfy the equation.

$$210\times 137^{\frac{1}{2}}+2454=4$$

Equation 140\sqrt{x}=4-8x has a unique solution.

$$x=\frac{1229-105\sqrt{137}}{8}$$