Skill: Estimate solutions using real numbers

To estimate a solution involving a square root, find two rational numbers that the solution lies between. Then narrow the interval and round to a reasonable decimal.

Example: Estimate the solutions of

$$x^2=20$$

First, recognize that the positive solution is $\sqrt{20}$.

Since

$$4^2=16 \quad\text{and}\quad 5^2=25,$$

we know

$$4<\sqrt{20}<5.$$

On a number line, $\sqrt{20}$ is between $4$ and $5$. To get a closer estimate, test tenths:

$$4.4^2=19.36$$

and

$$4.5^2=20.25.$$

Because $19.36<20<20.25$,

$$4.4<\sqrt{20}<4.5.$$

Now test hundredths:

$$4.47^2=19.9809$$

and

$$4.48^2=20.0704.$$

Therefore,

$$4.47<\sqrt{20}<4.48.$$

So, to the nearest hundredth,

$$\sqrt{20}\approx 4.47.$$

The equation $x^2=20$ has both a positive and a negative solution:

$$x=\pm\sqrt{20}\approx \pm 4.47.$$

The symbol $\approx$ means “approximately equal to.” The exact solutions are $\pm\sqrt{20}$, while $\pm4.47$ are decimal estimates.