# How do you use the square roots of 144 and 169 to estimate the square root of 154?

Apr 10, 2015

Since $\sqrt{144} = 12$ and $\sqrt{169} = 13$, $f \left(x\right) = {x}^{2}$ is quite steep by this point, and $\Delta x = 1$ is a small change in x, it is a reasonable assumption to estimate this portion of the graph as linear, and say that:

Difference:
$169 - 144 = 25$

Percent of the way through the difference:
$154 - 144 = 10$
=> 10/25 = 40%

Thus a nice estimate for the square root of 154 is 0.4 + 12 = 12.4. The true square root is 12.410, with a -0.0780% error.