Solve sqrt{x}+0.3=sqrt{x+4}

$$\sqrt { x } + 0.3 = \sqrt { x + 4 }$$

Solve for x

$$x = \frac{152881}{3600} = 42\frac{1681}{3600} \approx 42.466944444$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}+0.3\right)^{2}=\left(\sqrt{x+4}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x}+0.3\right)^{2}.

$$\left(\sqrt{x}\right)^{2}+0.6\sqrt{x}+0.09=\left(\sqrt{x+4}\right)^{2}$$

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

$$x+0.6\sqrt{x}+0.09=\left(\sqrt{x+4}\right)^{2}$$

Calculate \sqrt{x+4} to the power of 2 and get x+4.

$$x+0.6\sqrt{x}+0.09=x+4$$

Subtract x from both sides.

$$x+0.6\sqrt{x}+0.09-x=4$$

Combine x and -x to get 0.

$$0.6\sqrt{x}+0.09=4$$

Subtract 0.09 from both sides.

$$0.6\sqrt{x}=4-0.09$$

Subtract 0.09 from 4 to get 3.91.

$$0.6\sqrt{x}=3.91$$

Divide both sides by 0.6.

$$\sqrt{x}=\frac{3.91}{0.6}$$

Expand \frac{3.91}{0.6} by multiplying both numerator and the denominator by 100.

$$\sqrt{x}=\frac{391}{60}$$

Square both sides of the equation.

$$x=\frac{152881}{3600}$$

Substitute \frac{152881}{3600} for x in the equation \sqrt{x}+0.3=\sqrt{x+4}.

$$\sqrt{\frac{152881}{3600}}+0.3=\sqrt{\frac{152881}{3600}+4}$$

Simplify. The value x=\frac{152881}{3600} satisfies the equation.

$$\frac{409}{60}=\frac{409}{60}$$

Equation \sqrt{x}+0.3=\sqrt{x+4} has a unique solution.

$$x=\frac{152881}{3600}$$