Solve 30x-16sqrt{x}+2=0

$$30x-16 \sqrt{ x } +2=0$$

Solve for x

$$x=\frac{1}{9}\approx 0.111111111$$

$$x=\frac{1}{25}=0.04$$

Solution Steps

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$30x-16\sqrt{x}=-2$$

Subtract 30x from both sides of the equation.

$$-16\sqrt{x}=-2-30x$$

Square both sides of the equation.

$$\left(-16\sqrt{x}\right)^{2}=\left(-2-30x\right)^{2}$$

Expand \left(-16\sqrt{x}\right)^{2}.

$$\left(-16\right)^{2}\left(\sqrt{x}\right)^{2}=\left(-2-30x\right)^{2}$$

Calculate -16 to the power of 2 and get 256.

$$256\left(\sqrt{x}\right)^{2}=\left(-2-30x\right)^{2}$$

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

$$256x=\left(-2-30x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2-30x\right)^{2}.

$$256x=4+120x+900x^{2}$$

Subtract 120x from both sides.

$$256x-120x=4+900x^{2}$$

Combine 256x and -120x to get 136x.

$$136x=4+900x^{2}$$

Subtract 900x^{2} from both sides.

$$136x-900x^{2}=4$$

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.

$$-900x^{2}+136x=4$$

Subtract 4 from both sides of the equation.

$$-900x^{2}+136x-4=4-4$$

Subtracting 4 from itself leaves 0.

$$-900x^{2}+136x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -900 for a, 136 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-136±\sqrt{136^{2}-4\left(-900\right)\left(-4\right)}}{2\left(-900\right)}$$

Square 136.

$$x=\frac{-136±\sqrt{18496-4\left(-900\right)\left(-4\right)}}{2\left(-900\right)}$$

Multiply -4 times -900.

$$x=\frac{-136±\sqrt{18496+3600\left(-4\right)}}{2\left(-900\right)}$$

Multiply 3600 times -4.

$$x=\frac{-136±\sqrt{18496-14400}}{2\left(-900\right)}$$

Add 18496 to -14400.

$$x=\frac{-136±\sqrt{4096}}{2\left(-900\right)}$$

Take the square root of 4096.

$$x=\frac{-136±64}{2\left(-900\right)}$$

Multiply 2 times -900.

$$x=\frac{-136±64}{-1800}$$

Now solve the equation x=\frac{-136±64}{-1800} when ± is plus. Add -136 to 64.

$$x=-\frac{72}{-1800}$$

Reduce the fraction \frac{-72}{-1800} to lowest terms by extracting and canceling out 72.

$$x=\frac{1}{25}$$

Now solve the equation x=\frac{-136±64}{-1800} when ± is minus. Subtract 64 from -136.

$$x=-\frac{200}{-1800}$$

Reduce the fraction \frac{-200}{-1800} to lowest terms by extracting and canceling out 200.

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

The equation is now solved.

$$x=\frac{1}{25} x=\frac{1}{9}$$

Substitute \frac{1}{25} for x in the equation 30x-16\sqrt{x}+2=0.

$$30\times \frac{1}{25}-16\sqrt{\frac{1}{25}}+2=0$$

Simplify. The value x=\frac{1}{25} satisfies the equation.

$$0=0$$

Substitute \frac{1}{9} for x in the equation 30x-16\sqrt{x}+2=0.

$$30\times \frac{1}{9}-16\sqrt{\frac{1}{9}}+2=0$$

Simplify. The value x=\frac{1}{9} satisfies the equation.

$$0=0$$

List all solutions of -16\sqrt{x}=-30x-2.

$$x=\frac{1}{25} x=\frac{1}{9}$$