1Cademy - Solving $$\sqrt{m} + 1 = \sqrt{m + 9}$$

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Repeating the Isolate-and-Square Steps in Radical Equations

Example

Solving $\sqrt{m} + 1 = \sqrt{m + 9}$

Solve the radical equation $\sqrt{m} + 1 = \sqrt{m + 9}$ , which has square root expressions on both sides but one side contains a radical plus a constant — so squaring does not eliminate all radicals in a single step.

Step 1 — Identify the isolated radical. The right side already consists of a single square root, $\sqrt{m + 9}$ , so it is isolated. Square both sides:

$(\sqrt{m} + 1)^2 = (\sqrt{m + 9})^2$

Step 2 — Expand using the Binomial Squares Pattern. The left side is a binomial squared with $a = \sqrt{m}$ and $b = 1$ . Apply $(a + b)^2 = a^2 + 2ab + b^2$ :

$m + 2\sqrt{m} + 1 = m + 9$

A radical term $2\sqrt{m}$ still remains, so the isolate-and-square process must be repeated.

Step 3 — Isolate the remaining radical. Subtract $m$ and $1$ from both sides:

$2\sqrt{m} = 8$

Step 4 — Square both sides again:

$(2\sqrt{m})^2 = 8^2$

$4m = 64$

Step 5 — Solve the resulting linear equation. Divide both sides by $4$ :

$m = 16$

Step 6 — Check. Substituting $m = 16$ into the original equation confirms the solution (verification is left to the reader).

The solution is $m = 16$ . This example illustrates that when one side of a radical equation contains both a square root and a constant ( $\sqrt{m} + 1$ ), squaring that side via the Binomial Squares Pattern produces a middle term that still contains a radical, requiring a second round of isolating and squaring.

Updated 2026-04-21

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OpenStax Elementary Algebra

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